Oblique Central Impact

1

• Oblique Central Impact
• Motion is both along the line of impact and
• perpendicular to the line of impact

2

• Oblique Central Impact
• Motion is both along the line of impact and
• perpendicular to the line of impact
• Momentum is conserved along the line of impact
• mAvA1 mBvB1 mAvA2 mBvB2

3

• Oblique Central Impact
• Motion is both along the line of impact and
• perpendicular to the line of impact
• Momentum is conserved along the line of impact
• mAvA1 mBvB1 mAvA2 mBvB2
• No exchange of momentum perpendicular to line of
• Impact vA1 vA2 and vB1 vB2

4

• Oblique Central Impact
• Motion is both along the line of impact and
• perpendicular to the line of impact
• Momentum is conserved along the line of impact
• mAvA1 mBvB1 mAvA2 mBvB2
• No exchange of momentum perpendicular to line of
• Impact vA1 vA2 and vB1 vB2
• coefficient of restitution is the ratio of the
  relative
• velocity after impact to the relative velocity
  before
• impact along the line of impact
• e - (vA2 - vB2)/(vA1 - vB1)

5

• Procedure for Impact Problems
• Step1 Determine the velocity of both objects just
  before and just after
• impact along and perpendicular to the
  line of impact either from
• the problem or from another
  calculation.
• Step 2 Apply conservation momentum to just before
  to just after
• Impact along the line of impact. Note
  if one of the objects is
• the earth omit this step.
• Step 3 Apply the coefficient of restitution
  equation to just before to just
• after Impact along the line of impact.
• Step solve for the unknowns.

6

• Problem 13.166 Two identical hockey pucks are
• moving on a hockey rink at the same speed of 10
• ft/s in parallel and opposite directions when
  they
• strike each other as shown. Assuming a
• coefficient of restitution e 1 , determine the
• magnitude and direction of the velocity of each
• puck after impact.

7
y

• Problem 13.166 Two identical hockey pucks are
• moving on a hockey rink at the same speed of 10
• ft/s in parallel and opposite directions when
  they
• strike each other as shown. Assuming a
• coefficient of restitution e 1 , determine the
• magnitude and direction of the velocity of each
• puck after impact.
• First determine the line of impact

x
8
y

• Problem 13.166 Two identical hockey pucks are
• moving on a hockey rink at the same speed of 10
• ft/s in parallel and opposite directions when
  they
• strike each other as shown. Assuming a
• coefficient of restitution e 1 , determine the
• magnitude and direction of the velocity of each
• puck after impact.
• First determine the line of impact and resolve
  the
• pre-impact velocities into components along and
• perpendicular to the line of impact.
• vA vAcos20i vAsin20j

x
9
y

• Problem 13.166 Two identical hockey pucks are
• moving on a hockey rink at the same speed of 10
• ft/s in parallel and opposite directions when
  they
• strike each other as shown. Assuming a
• coefficient of restitution e 1 , determine the
• magnitude and direction of the velocity of each
• puck after impact.
• First determine the line of impact and resolve
  the
• pre-impact velocities into components along and
• perpendicular to the line of impact.
• vA vAcos20i vAsin20j 9.40i 3.42j

x
10
y

• Problem 13.166 Two identical hockey pucks are
• moving on a hockey rink at the same speed of 10
• ft/s in parallel and opposite directions when
  they
• strike each other as shown. Assuming a
• coefficient of restitution e 1 , determine the
• magnitude and direction of the velocity of each
• puck after impact.
• First determine the line of impact and resolve
  the
• pre-impact velocities into components along and
• perpendicular to the line of impact.
• vA vAcos20i vAsin20j 9.40i 3.42j
• vB - vBcos20i vBsin20j

x
11
y

• Problem 13.166 Two identical hockey pucks are
• moving on a hockey rink at the same speed of 10
• ft/s in parallel and opposite directions when
  they
• strike each other as shown. Assuming a
• coefficient of restitution e 1 , determine the
• magnitude and direction of the velocity of each
• puck after impact.
• First determine the line of impact and resolve
  the
• pre-impact velocities into components along and
• perpendicular to the line of impact.
• vA vAcos20i vAsin20j 9.40i 3.42j
• vB - vBcos20i vBsin20j - 9.40i 3.42j

x
12
y

• Problem 13.166 Two identical hockey pucks are
• moving on a hockey rink at the same speed of 10
• ft/s in parallel and opposite directions when
  they
• strike each other as shown. Assuming a
• coefficient of restitution e 1 , determine the
• magnitude and direction of the velocity of each
• puck after impact.
• First determine the line of impact and resolve
  the
• pre-impact velocities into components along and
• perpendicular to the line of impact.
• vA vAcos20i vAsin20j 9.40i 3.42j
• vB - vBcos20i vBsin20j - 9.40i 3.42j
• along x-axis S(mv)1 S(mv)2

x
13
y

• Problem 13.166 Two identical hockey pucks are
• moving on a hockey rink at the same speed of 10
• ft/s in parallel and opposite directions when
  they
• strike each other as shown. Assuming a
• coefficient of restitution e 1 , determine the
• magnitude and direction of the velocity of each
• puck after impact.
• First determine the line of impact and resolve
  the
• pre-impact velocities into components along and
• perpendicular to the line of impact.
• vA vAcos20i vAsin20j 9.40i 3.42j
• vB - vBcos20i vBsin20j - 9.40i 3.42j
• along x-axis S(mv)1 S(mv)2
• m9.40 m(-9.40)

x
14
y

• Problem 13.166 Two identical hockey pucks are
• moving on a hockey rink at the same speed of 10
• ft/s in parallel and opposite directions when
  they
• strike each other as shown. Assuming a
• coefficient of restitution e 1 , determine the
• magnitude and direction of the velocity of each
• puck after impact.
• First determine the line of impact and resolve
  the
• pre-impact velocities into components along and
• perpendicular to the line of impact.
• vA vAcos20i vAsin20j 9.40i 3.42j
• vB - vBcos20i vBsin20j - 9.40i 3.42j
• along x-axis S(mv)1 S(mv)2
• m9.40 m(-9.40) mvA2X mvB2X

x
15
y

• Problem 13.166 Two identical hockey pucks are
• moving on a hockey rink at the same speed of 10
• ft/s in parallel and opposite directions when
  they
• strike each other as shown. Assuming a
• coefficient of restitution e 1 , determine the
• magnitude and direction of the velocity of each
• puck after impact.
• First determine the line of impact and resolve
  the
• pre-impact velocities into components along and
• perpendicular to the line of impact.
• vA vAcos20i vAsin20j 9.40i 3.42j
• vB - vBcos20i vBsin20j - 9.40i 3.42j
• along x-axis S(mv)1 S(mv)2
• m9.40 m(-9.40) mvA2X mvB2X
• 0 vA2X vB2X

x
16
y

• Problem 13.166 Two identical hockey pucks are
• moving on a hockey rink at the same speed of 10
• ft/s in parallel and opposite directions when
  they
• strike each other as shown. Assuming a
• coefficient of restitution e 1 , determine the
• magnitude and direction of the velocity of each
• puck after impact.
• First determine the line of impact and resolve
  the
• pre-impact velocities into components along and
• perpendicular to the line of impact.
• vA vAcos20i vAsin20j 9.40i 3.42j
• vB - vBcos20i vBsin20j - 9.40i 3.42j
• along x-axis S(mv)1 S(mv)2
• m9.40 m(-9.40) mvA2X mvB2X
• 0 vA2X vB2X
• e - (vA2X - vB2X)/(vA1X - vB1X)

x
17
y

• Problem 13.166 Two identical hockey pucks are
• moving on a hockey rink at the same speed of 10
• ft/s in parallel and opposite directions when
  they
• strike each other as shown. Assuming a
• coefficient of restitution e 1 , determine the
• magnitude and direction of the velocity of each
• puck after impact.
• First determine the line of impact and resolve
  the
• pre-impact velocities into components along and
• perpendicular to the line of impact.
• vA vAcos20i vAsin20j 9.40i 3.42j
• vB - vBcos20i vBsin20j - 9.40i 3.42j
• along x-axis S(mv)1 S(mv)2
• m9.40 m(-9.40) mvA2X mvB2X
• 0 vA2X vB2X
• e - (vA2X - vB2X)/(vA1X - vB1X)
• 1 - (vA2X - vB2X)

x
18
y

• Problem 13.166 Two identical hockey pucks are
• moving on a hockey rink at the same speed of 10
• ft/s in parallel and opposite directions when
  they
• strike each other as shown. Assuming a
• coefficient of restitution e 1 , determine the
• magnitude and direction of the velocity of each
• puck after impact.
• First determine the line of impact and resolve
  the
• pre-impact velocities into components along and
• perpendicular to the line of impact.
• vA vAcos20i vAsin20j 9.40i 3.42j
• vB - vBcos20i vBsin20j - 9.40i 3.42j
• along x-axis S(mv)1 S(mv)2
• m9.40 m(-9.40) mvA2X mvB2X
• 0 vA2X vB2X
• e - (vA2X - vB2X)/(vA1X - vB1X)
• 1 - (vA2X - vB2X)/(9.40 (-9.40))

x
19
y

• Problem 13.166 Two identical hockey pucks are
• moving on a hockey rink at the same speed of 10
• ft/s in parallel and opposite directions when
  they
• strike each other as shown. Assuming a
• coefficient of restitution e 1 , determine the
• magnitude and direction of the velocity of each
• puck after impact.
• First determine the line of impact and resolve
  the
• pre-impact velocities into components along and
• perpendicular to the line of impact.
• vA vAcos20i vAsin20j 9.40i 3.42j
• vB - vBcos20i vBsin20j - 9.40i 3.42j
• along x-axis S(mv)1 S(mv)2
• m9.40 m(-9.40) mvA2X mvB2X
• 0 vA2X vB2X
• e - (vA2X - vB2X)/(vA1X - vB1X)
• 1 - (vA2X - vB2X)/(9.40 (-9.40))
• - 18.80 vA2X - vB2X

x
20
y

• Problem 13.166 Two identical hockey pucks are
• moving on a hockey rink at the same speed of 10
• ft/s in parallel and opposite directions when
  they
• strike each other as shown. Assuming a
• coefficient of restitution e 1 , determine the
• magnitude and direction of the velocity of each
• puck after impact.
• First determine the line of impact and resolve
  the
• pre-impact velocities into components along and
• perpendicular to the line of impact.
• vA vAcos20i vAsin20j 9.40i 3.42j
• vB - vBcos20i vBsin20j - 9.40i 3.42j
• along x-axis S(mv)1 S(mv)2
• m9.40 m(-9.40) mvA2X mvB2X
• 0 vA2X vB2X
• e - (vA2X - vB2X)/(vA1X - vB1X)
• 1 - (vA2X - vB2X)/(9.40 (-9.40))
• -18.80 vA2 - vB2
• 2 equations 2 unknowns

x
21
y

• Problem 13.166 Two identical hockey pucks are
• moving on a hockey rink at the same speed of 10
• ft/s in parallel and opposite directions when
  they
• strike each other as shown. Assuming a
• coefficient of restitution e 1 , determine the
• magnitude and direction of the velocity of each
• puck after impact.
• First determine the line of impact and resolve
  the
• pre-impact velocities into components along and
• perpendicular to the line of impact.
• vA vAcos20i vAsin20j 9.40i 3.42j
• vB - vBcos20i vBsin20j - 9.40i 3.42j
• along x-axis S(mv)1 S(mv)2
• m9.40 m(-9.40) mvA2X mvB2X
• 0 vA2X vB2X
• e - (vA2X - vB2X)/(vA1X - vB1X)
• 1 - (vA2X - vB2X)/(9.40 (-9.40))
• -18.80 vA2 - vB2
• 2 equations 2 unknowns

x
22
y

• Problem 13.166 Two identical hockey pucks are
• moving on a hockey rink at the same speed of 10
• ft/s in parallel and opposite directions when
  they
• strike each other as shown. Assuming a
• coefficient of restitution e 1 , determine the
• magnitude and direction of the velocity of each
• puck after impact.
• First determine the line of impact and resolve
  the
• pre-impact velocities into components along and
• perpendicular to the line of impact.
• vA vAcos20i vAsin20j 9.40i 3.42j
• vB - vBcos20i vBsin20j - 9.40i 3.42j
• along x-axis S(mv)1 S(mv)2
• m9.40 m(-9.40) mvA2X mvB2X
• 0 vA2X vB2X
• e - (vA2X - vB2X)/(vA1X - vB1X)
• 1 - (vA2X - vB2X)/(9.40 (-9.40))
• -18.80 vA2 - vB2
• 2 equations 2 unknowns

x
200
23

• Problem involving combinations of Work Energy,
• Impulse Momentum, and Impact
• Problem 13.178 A 2.5 lb block B is moving with a
• velocity of magnitude v0 6 ft/s as it hits a
  1.5 lb
• sphere A, which is at rest and hanging from a
  cord
• attached at O. Knowing that mk 0.6 between the
  block
• and the horizontal surface and e 0.8 between
  the
• block and the sphere, determine after impact, (a)
  the
• maximum height h reached by the sphere, (b) the
• distance x traveled by the block.

24

• Problem involving combinations of Work Energy,
• Impulse Momentum, and Impact
• Problem 13.178 A 2.5 lb block B is moving with a
• velocity of magnitude v0 6 ft/s as it hits a
  1.5 lb
• sphere A, which is at rest and hanging from a
  cord
• attached at O. Knowing that mk 0.6 between the
  block
• and the horizontal surface and e 0.8 between
  the
• block and the sphere, determine after impact, (a)
  the
• maximum height h reached by the sphere, (b) the
• distance x traveled by the block.
• First we have the impact between the block and
  the
• sphere momentum conserved along the line of
  impact
• S(mv)1 S(mv)2

25

• Problem involving combinations of Work Energy,
• Impulse Momentum, and Impact
• Problem 13.178 A 2.5 lb block B is moving with a
• velocity of magnitude v0 6 ft/s as it hits a
  1.5 lb
• sphere A, which is at rest and hanging from a
  cord
• attached at O. Knowing that mk 0.6 between the
  block
• and the horizontal surface and e 0.8 between
  the
• block and the sphere, determine after impact, (a)
  the
• maximum height h reached by the sphere, (b) the
• distance x traveled by the block.
• First we have the impact between the block and
  the
• sphere momentum conserved along the line of
  impact
• S(mv)1 S(mv)2
• (2.5/g)(-6) (1.5/g)0

26

• Problem involving combinations of Work Energy,
• Impulse Momentum, and Impact
• Problem 13.178 A 2.5 lb block B is moving with a
• velocity of magnitude v0 6 ft/s as it hits a
  1.5 lb
• sphere A, which is at rest and hanging from a
  cord
• attached at O. Knowing that mk 0.6 between the
  block
• and the horizontal surface and e 0.8 between
  the
• block and the sphere, determine after impact, (a)
  the
• maximum height h reached by the sphere, (b) the
• distance x traveled by the block.
• First we have the impact between the block and
  the
• sphere momentum conserved along the line of
  impact
• S(mv)1 S(mv)2
• (2.5/g)(-6) (1.5/g)0 (2.5/g)vBX2 (1.5/g)vSX2

27

• Problem involving combinations of Work Energy,
• Impulse Momentum, and Impact
• Problem 13.178 A 2.5 lb block B is moving with a
• velocity of magnitude v0 6 ft/s as it hits a
  1.5 lb
• sphere A, which is at rest and hanging from a
  cord
• attached at O. Knowing that mk 0.6 between the
  block
• and the horizontal surface and e 0.8 between
  the
• block and the sphere, determine after impact, (a)
  the
• maximum height h reached by the sphere, (b) the
• distance x traveled by the block.
• First we have the impact between the block and
  the
• sphere momentum conserved along the line of
  impact
• S(mv)1 S(mv)2
• (2.5/g)(-6) (1.5/g)0 (2.5/g)vBX2 (1.5/g)vSX2
• e - (vBX2 - vSX2)/(vBX1 - vSX1)

28

• Problem involving combinations of Work Energy,
• Impulse Momentum, and Impact
• Problem 13.178 A 2.5 lb block B is moving with a
• velocity of magnitude v0 6 ft/s as it hits a
  1.5 lb
• sphere A, which is at rest and hanging from a
  cord
• attached at O. Knowing that mk 0.6 between the
  block
• and the horizontal surface and e 0.8 between
  the
• block and the sphere, determine after impact, (a)
  the
• maximum height h reached by the sphere, (b) the
• distance x traveled by the block.
• First we have the impact between the block and
  the
• sphere momentum conserved along the line of
  impact
• S(mv)1 S(mv)2
• (2.5/g)(-6) (1.5/g)0 (2.5/g)vBX2 (1.5/g)vSX2
• e - (vBX2 - vSX2)/(vBX1 - vSX1)
• 0.8 - (vBX2 - vSX2)

29

• Problem involving combinations of Work Energy,
• Impulse Momentum, and Impact
• Problem 13.178 A 2.5 lb block B is moving with a
• velocity of magnitude v0 6 ft/s as it hits a
  1.5 lb
• sphere A, which is at rest and hanging from a
  cord
• attached at O. Knowing that mk 0.6 between the
  block
• and the horizontal surface and e 0.8 between
  the
• block and the sphere, determine after impact, (a)
  the
• maximum height h reached by the sphere, (b) the
• distance x traveled by the block.
• First we have the impact between the block and
  the
• sphere momentum conserved along the line of
  impact
• S(mv)1 S(mv)2
• (2.5/g)(-6) (1.5/g)0 (2.5/g)vBX2 (1.5/g)vSX2
• e - (vBX2 - vSX2)/(vBX1 - vSX1)
• 0.8 - (vBX2 - vSX2)/(-6 0)

30

• Problem involving combinations of Work Energy,
• Impulse Momentum, and Impact
• Problem 13.178 A 2.5 lb block B is moving with a
• velocity of magnitude v0 6 ft/s as it hits a
  1.5 lb
• sphere A, which is at rest and hanging from a
  cord
• attached at O. Knowing that mk 0.6 between the
  block
• and the horizontal surface and e 0.8 between
  the
• block and the sphere, determine after impact, (a)
  the
• maximum height h reached by the sphere, (b) the
• distance x traveled by the block.
• First we have the impact between the block and
  the
• sphere momentum conserved along the line of
  impact
• S(mv)1 S(mv)2
• (2.5/g)(-6) (1.5/g)0 (2.5/g)vBX2 (1.5/g)vSX2
• e - (vBX2 - vSX2)/(vBX1 - vSX1)
• 0.8 - (vBX2 - vSX2)/(-6 0)
• vBX2 -1.95 ft/s vSX2 -6.75 ft/s

31

• Problem involving combinations of Work Energy,
• Impulse Momentum, and Impact
• Problem 13.178 A 2.5 lb block B is moving with a
• velocity of magnitude v0 6 ft/s as it hits a
  1.5 lb
• sphere A, which is at rest and hanging from a
  cord
• attached at O. Knowing that mk 0.6 between the
  block
• and the horizontal surface and e 0.8 between
  the
• block and the sphere, determine after impact, (a)
  the
• maximum height h reached by the sphere, (b) the
• distance x traveled by the block.
• First we have the impact between the block and
  the
• sphere momentum conserved along the line of
  impact
• vBX2 -1.95 ft/s vSX2 -6.75 ft/s
• Sphere by itself from just after impact to
  highest point
• Use conservation of energy T2 V2g T3
  V3g

Datum
32

• Problem involving combinations of Work Energy,
• Impulse Momentum, and Impact
• Problem 13.178 A 2.5 lb block B is moving with a
• velocity of magnitude v0 6 ft/s as it hits a
  1.5 lb
• sphere A, which is at rest and hanging from a
  cord
• attached at O. Knowing that mk 0.6 between the
  block
• and the horizontal surface and e 0.8 between
  the
• block and the sphere, determine after impact, (a)
  the
• maximum height h reached by the sphere, (b) the
• distance x traveled by the block.
• First we have the impact between the block and
  the
• sphere momentum conserved along the line of
  impact
• vBX2 -1.95 ft/s vSX2 -6.75 ft/s
• Sphere by itself from just after impact to
  highest point
• Use conservation of energy T2 V2g T3
  V3g
• (1/2)(1.5/g)(-6.75)2

Datum
33

• Problem involving combinations of Work Energy,
• Impulse Momentum, and Impact
• Problem 13.178 A 2.5 lb block B is moving with a
• velocity of magnitude v0 6 ft/s as it hits a
  1.5 lb
• sphere A, which is at rest and hanging from a
  cord
• attached at O. Knowing that mk 0.6 between the
  block
• and the horizontal surface and e 0.8 between
  the
• block and the sphere, determine after impact, (a)
  the
• maximum height h reached by the sphere, (b) the
• distance x traveled by the block.
• First we have the impact between the block and
  the
• sphere momentum conserved along the line of
  impact
• vBX2 -1.95 ft/s vSX2 -6.75 ft/s
• Sphere by itself from just after impact to
  highest point
• Use conservation of energy T2 V2g T3
  V3g
• (1/2)(1.5/g)(-6.75)2 1.5(0)

Datum
34

• Problem involving combinations of Work Energy,
• Impulse Momentum, and Impact
• Problem 13.178 A 2.5 lb block B is moving with a
• velocity of magnitude v0 6 ft/s as it hits a
  1.5 lb
• sphere A, which is at rest and hanging from a
  cord
• attached at O. Knowing that mk 0.6 between the
  block
• and the horizontal surface and e 0.8 between
  the
• block and the sphere, determine after impact, (a)
  the
• maximum height h reached by the sphere, (b) the
• distance x traveled by the block.
• First we have the impact between the block and
  the
• sphere momentum conserved along the line of
  impact
• vBX2 -1.95 ft/s vSX2 -6.75 ft/s
• Sphere by itself from just after impact to
  highest point
• Use conservation of energy T2 V2g T3
  V3g
• (1/2)(1.5/g)(-6.75)2 1.5(0) (1/2)(1.5/g)02

Datum
35

• Problem involving combinations of Work Energy,
• Impulse Momentum, and Impact
• Problem 13.178 A 2.5 lb block B is moving with a
• velocity of magnitude v0 6 ft/s as it hits a
  1.5 lb
• sphere A, which is at rest and hanging from a
  cord
• attached at O. Knowing that mk 0.6 between the
  block
• and the horizontal surface and e 0.8 between
  the
• block and the sphere, determine after impact, (a)
  the
• maximum height h reached by the sphere, (b) the
• distance x traveled by the block.
• First we have the impact between the block and
  the
• sphere momentum conserved along the line of
  impact
• vBX2 -1.95 ft/s vSX2 -6.75 ft/s
• Sphere by itself from just after impact to
  highest point
• Use conservation of energy T2 V2g T3
  V3g
• (1/2)(1.5/g)(-6.75)2 1.5(0) (1/2)(1.5/g)02
  1.5h

Datum
36

• Problem involving combinations of Work Energy,
• Impulse Momentum, and Impact
• Problem 13.178 A 2.5 lb block B is moving with a
• velocity of magnitude v0 6 ft/s as it hits a
  1.5 lb
• sphere A, which is at rest and hanging from a
  cord
• attached at O. Knowing that mk 0.6 between the
  block
• and the horizontal surface and e 0.8 between
  the
• block and the sphere, determine after impact, (a)
  the
• maximum height h reached by the sphere, (b) the
• distance x traveled by the block.
• First we have the impact between the block and
  the
• sphere momentum conserved along the line of
  impact
• vBX2 -1.95 ft/s vSX2 -6.75 ft/s
• Sphere by itself from just after impact to
  highest point
• Use conservation of energy T2 V2g T3
  V3g
• (1/2)(1.5/g)(-6.75)2 1.5(0) (1/2)(1.5/g)02
  1.5h
• h 0.707 ft

Datum
37

• Problem involving combinations of Work Energy,
• Impulse Momentum, and Impact
• Problem 13.178 A 2.5 lb block B is moving with a
• velocity of magnitude v0 6 ft/s as it hits a
  1.5 lb
• sphere A, which is at rest and hanging from a
  cord
• attached at O. Knowing that mk 0.6 between the
  block
• and the horizontal surface and e 0.8 between
  the
• block and the sphere, determine after impact, (a)
  the
• maximum height h reached by the sphere, (b) the
• distance x traveled by the block.
• First we have the impact between the block and
  the
• sphere momentum conserved along the line of
  impact
• vBX2 -1.95 ft/s vSX2 -6.75 ft/s
• h 0.707 ft
• Block by itself after impact has friction use
  work energy
• TB2 U23 TB3

Datum
y
x
0.6N
N
2.5
38

• Problem involving combinations of Work Energy,
• Impulse Momentum, and Impact
• Problem 13.178 A 2.5 lb block B is moving with a
• velocity of magnitude v0 6 ft/s as it hits a
  1.5 lb
• sphere A, which is at rest and hanging from a
  cord
• attached at O. Knowing that mk 0.6 between the
  block
• and the horizontal surface and e 0.8 between
  the
• block and the sphere, determine after impact, (a)
  the
• maximum height h reached by the sphere, (b) the
• distance x traveled by the block.
• First we have the impact between the block and
  the
• sphere momentum conserved along the line of
  impact
• vBX2 -1.95 ft/s vSX2 -6.75 ft/s
• h 0.707 ft
• Block by itself after impact has friction use
  work energy
• TB2 U23 TB3
• (1/2)(2.5/g)(-1.95)2

Datum
y
x
0.6N
N
2.5
39

• Problem involving combinations of Work Energy,
• Impulse Momentum, and Impact
• Problem 13.178 A 2.5 lb block B is moving with a
• velocity of magnitude v0 6 ft/s as it hits a
  1.5 lb
• sphere A, which is at rest and hanging from a
  cord
• attached at O. Knowing that mk 0.6 between the
  block
• and the horizontal surface and e 0.8 between
  the
• block and the sphere, determine after impact, (a)
  the
• maximum height h reached by the sphere, (b) the
• distance x traveled by the block.
• First we have the impact between the block and
  the
• sphere momentum conserved along the line of
  impact
• vBX2 -1.95 ft/s vSX2 -6.75 ft/s
• h 0.707 ft
• Block by itself after impact has friction use
  work energy
• TB2 U23 TB3
• (1/2)(2.5/g)(-1.95)2 0.6Nx

Datum
y
x
0.6N
N
2.5
40

• Problem involving combinations of Work Energy,
• Impulse Momentum, and Impact
• Problem 13.178 A 2.5 lb block B is moving with a
• velocity of magnitude v0 6 ft/s as it hits a
  1.5 lb
• sphere A, which is at rest and hanging from a
  cord
• attached at O. Knowing that mk 0.6 between the
  block
• and the horizontal surface and e 0.8 between
  the
• block and the sphere, determine after impact, (a)
  the
• maximum height h reached by the sphere, (b) the
• distance x traveled by the block.
• First we have the impact between the block and
  the
• sphere momentum conserved along the line of
  impact
• vBX2 -1.95 ft/s vSX2 -6.75 ft/s
• h 0.707 ft
• Block by itself after impact has friction use
  work energy
• TB2 U23 TB3
• (1/2)(2.5/g)(-1.95)2 0.6Nx (1/2)(2.5/g)02

Datum
y
x
0.6N
N
2.5
41

• Problem involving combinations of Work Energy,
• Impulse Momentum, and Impact
• Problem 13.178 A 2.5 lb block B is moving with a
• velocity of magnitude v0 6 ft/s as it hits a
  1.5 lb
• sphere A, which is at rest and hanging from a
  cord
• attached at O. Knowing that mk 0.6 between the
  block
• and the horizontal surface and e 0.8 between
  the
• block and the sphere, determine after impact, (a)
  the
• maximum height h reached by the sphere, (b) the
• distance x traveled by the block.
• First we have the impact between the block and
  the
• sphere momentum conserved along the line of
  impact
• vBX2 -1.95 ft/s vSX2 -6.75 ft/s
• h 0.707 ft
• Block by itself after impact has friction use
  work energy
• TB2 U23 TB3
• (1/2)(2.5/g)(-1.95)2 0.6Nx (1/2)(2.5/g)02
• SFY maY ? N 2.5 (2.5/g)0

Datum
y
x
0.6N
N
2.5
42

• Problem involving combinations of Work Energy,
• Impulse Momentum, and Impact
• Problem 13.178 A 2.5 lb block B is moving with a
• velocity of magnitude v0 6 ft/s as it hits a
  1.5 lb
• sphere A, which is at rest and hanging from a
  cord
• attached at O. Knowing that mk 0.6 between the
  block
• and the horizontal surface and e 0.8 between
  the
• block and the sphere, determine after impact, (a)
  the
• maximum height h reached by the sphere, (b) the
• distance x traveled by the block.
• First we have the impact between the block and
  the
• sphere momentum conserved along the line of
  impact
• vBX2 -1.95 ft/s vSX2 -6.75 ft/s
• h 0.707 ft
• Block by itself after impact has friction use
  work energy
• TB2 U23 TB3
• (1/2)(2.5/g)(-1.95)2 0.6Nx (1/2)(2.5/g)02
• SFY maY ? N 2.5 (2.5/g)0 ? N 2.5
• x - 0.0984 ft

Datum
y
x
0.6N
N
2.5
43

• Problem 13.188 A 2 kg sphere A strikes the
  frictionless
• inclined surface of a 6 kg wedge B at a 900 angle
  with a
• velocity of magnitude 4 m/s. The wedge can roll
  freely
• on the ground and is initially at rest. Knowing
  that the
• coefficient of restitution between the wedge and
  the
• sphere is 0.50 and that the inclined surface of
  the
• wedge forms an angle of 400 with the horizontal,
• determine (a) the velocities of the sphere and
  wedge
• immediately after impact, (b) the energy lost due
  to
• impact.

44

• Problem 13.188 A 2 kg sphere A strikes the
  frictionless
• inclined surface of a 6 kg wedge B at a 900 angle
  with a
• velocity of magnitude 4 m/s. The wedge can roll
  freely
• on the ground and is initially at rest. Knowing
  that the
• coefficient of restitution between the wedge and
  the
• sphere is 0.50 and that the inclined surface of
  the
• wedge forms an angle of 400 with the horizontal,
• determine (a) the velocities of the sphere and
  wedge
• immediately after impact, (b) the energy lost due
  to
• impact.

N
T
y
x
45

• Problem 13.188 A 2 kg sphere A strikes the
  frictionless
• inclined surface of a 6 kg wedge B at a 900 angle
  with a
• velocity of magnitude 4 m/s. The wedge can roll
  freely
• on the ground and is initially at rest. Knowing
  that the
• coefficient of restitution between the wedge and
  the
• sphere is 0.50 and that the inclined surface of
  the
• wedge forms an angle of 400 with the horizontal,
• determine (a) the velocities of the sphere and
  wedge
• immediately after impact, (b) the energy lost due
  to
• impact. For coefficient of restitution we use the
  T axis
• e - (vAT2 vBT2)/(vAT1 vBT1)

N
T
y
x
46

• Problem 13.188 A 2 kg sphere A strikes the
  frictionless
• inclined surface of a 6 kg wedge B at a 900 angle
  with a
• velocity of magnitude 4 m/s. The wedge can roll
  freely
• on the ground and is initially at rest. Knowing
  that the
• coefficient of restitution between the wedge and
  the
• sphere is 0.50 and that the inclined surface of
  the
• wedge forms an angle of 400 with the horizontal,
• determine (a) the velocities of the sphere and
  wedge
• immediately after impact, (b) the energy lost due
  to
• impact. For coefficient of restitution we use the
  T axis
• e - (vAT2 vBT2)/(vAT1 vBT1)
• 0.5 - (vAT2 vBT2)

N
T
y
x
47

• Problem 13.188 A 2 kg sphere A strikes the
  frictionless
• inclined surface of a 6 kg wedge B at a 900 angle
  with a
• velocity of magnitude 4 m/s. The wedge can roll
  freely
• on the ground and is initially at rest. Knowing
  that the
• coefficient of restitution between the wedge and
  the
• sphere is 0.50 and that the inclined surface of
  the
• wedge forms an angle of 400 with the horizontal,
• determine (a) the velocities of the sphere and
  wedge
• immediately after impact, (b) the energy lost due
  to
• impact. For coefficient of restitution we use the
  T axis
• e - (vAT2 vBT2)/(vAT1 vBT1)
• 0.5 - (vAT2 vBT2)/(-4 0)

N
T
y
x
48

• Problem 13.188 A 2 kg sphere A strikes the
  frictionless
• inclined surface of a 6 kg wedge B at a 900 angle
  with a
• velocity of magnitude 4 m/s. The wedge can roll
  freely
• on the ground and is initially at rest. Knowing
  that the
• coefficient of restitution between the wedge and
  the
• sphere is 0.50 and that the inclined surface of
  the
• wedge forms an angle of 400 with the horizontal,
• determine (a) the velocities of the sphere and
  wedge
• immediately after impact, (b) the energy lost due
  to
• impact. For coefficient of restitution we use the
  T axis
• e - (vAT2 vBT2)/(vAT1 vBT1)
• 0.5 - (vAT2 vBT2)/(-4 0)
• Since there is an unknown impulse in the y
  direction
• Use x direction for conservation of momentum
• S(mv)X1 S(mv)X2

N
T
y
x
49

• Problem 13.188 A 2 kg sphere A strikes the
  frictionless
• inclined surface of a 6 kg wedge B at a 900 angle
  with a
• velocity of magnitude 4 m/s. The wedge can roll
  freely
• on the ground and is initially at rest. Knowing
  that the
• coefficient of restitution between the wedge and
  the
• sphere is 0.50 and that the inclined surface of
  the
• wedge forms an angle of 400 with the horizontal,
• determine (a) the velocities of the sphere and
  wedge
• immediately after impact, (b) the energy lost due
  to
• impact. For coefficient of restitution we use the
  T axis
• e - (vAT2 vBT2)/(vAT1 vBT1)
• 0.5 - (vAT2 vBT2)/(-4 0)
• Since there is an unknown impulse in the y
  direction
• Use x direction for conservation of momentum
• S(mv)X1 S(mv)X2
• 2(4cos50) 6(0)

N
T
y
x
50

• Problem 13.188 A 2 kg sphere A strikes the
  frictionless
• inclined surface of a 6 kg wedge B at a 900 angle
  with a
• velocity of magnitude 4 m/s. The wedge can roll
  freely
• on the ground and is initially at rest. Knowing
  that the
• coefficient of restitution between the wedge and
  the
• sphere is 0.50 and that the inclined surface of
  the
• wedge forms an angle of 400 with the horizontal,
• determine (a) the velocities of the sphere and
  wedge
• immediately after impact, (b) the energy lost due
  to
• impact. For coefficient of restitution we use the
  T axis
• e - (vAT2 vBT2)/(vAT1 vBT1)
• 0.5 - (vAT2 vBT2)/(-4 0)
• Since there is an unknown impulse in the y
  direction
• Use x direction for conservation of momentum
• S(mv)X1 S(mv)X2
• 2(4cos50) 6(0) 2(-vAX2) 6vBX2

N
T
y
x
51

• Problem 13.188 A 2 kg sphere A strikes the
  frictionless
• inclined surface of a 6 kg wedge B at a 900 angle
  with a
• velocity of magnitude 4 m/s. The wedge can roll
  freely
• on the ground and is initially at rest. Knowing
  that the
• coefficient of restitution between the wedge and
  the
• sphere is 0.50 and that the inclined surface of
  the
• wedge forms an angle of 400 with the horizontal,
• determine (a) the velocities of the sphere and
  wedge
• immediately after impact, (b) the energy lost due
  to
• impact. For coefficient of restitution we use the
  T axis
• e - (vAT2 vBT2)/(vAT1 vBT1)
• 0.5 - (vAT2 vBT2)/(-4 0)
• Since there is an unknown impulse in the y
  direction
• Use x direction for conservation of momentum
• S(mv)X1 S(mv)X2
• 2(4cos50) 6(0) 2(-vAX2) 6vBX2
• -vAX2 vAT2cos50

N
T
y
x
52

• Problem 13.188 A 2 kg sphere A strikes the
  frictionless
• inclined surface of a 6 kg wedge B at a 900 angle
  with a
• velocity of magnitude 4 m/s. The wedge can roll
  freely
• on the ground and is initially at rest. Knowing
  that the
• coefficient of restitution between the wedge and
  the
• sphere is 0.50 and that the inclined surface of
  the
• wedge forms an angle of 400 with the horizontal,
• determine (a) the velocities of the sphere and
  wedge
• immediately after impact, (b) the energy lost due
  to
• impact. For coefficient of restitution we use the
  T axis
• e - (vAT2 vBT2)/(vAT1 vBT1)
• 0.5 - (vAT2 vBT2)/(-4 0)
• Since there is an unknown impulse in the y
  direction
• Use x direction for conservation of momentum
• S(mv)X1 S(mv)X2
• 2(4cos50) 6(0) 2(-vAX2) 6vBX2
• -vAX2 vAT2cos50 and vBT2 vBX2cos50

N
T
y
x
53

• Problem 13.188 A 2 kg sphere A strikes the
  frictionless
• inclined surface of a 6 kg wedge B at a 900 angle
  with a
• velocity of magnitude 4 m/s. The wedge can roll
  freely
• on the ground and is initially at rest. Knowing
  that the
• coefficient of restitution between the wedge and
  the
• sphere is 0.50 and that the inclined surface of
  the
• wedge forms an angle of 400 with the horizontal,
• determine (a) the velocities of the sphere and
  wedge
• immediately after impact, (b) the energy lost due
  to
• impact. For coefficient of restitution we use the
  T axis
• e - (vAT2 vBT2)/(vAT1 vBT1)
• 0.5 - (vAT2 vBT2)/(-4 0)
• Since there is an unknown impulse in the y
  direction
• Use x direction for conservation of momentum
• S(mv)X1 S(mv)X2
• 2(4cos50) 6(0) 2(-vAX2) 6vBX2
• -vAX2 vAT2cos50 and vBT2 vBX2cos50
• 4 equations 4 unknowns vAX2 , vAT2 , vBT2 , and
  vBX2

N
T
y
x
54

• Problem 13.188 A 2 kg sphere A strikes the
  frictionless
• inclined surface of a 6 kg wedge B at a 900 angle
  with a
• velocity of magnitude 4 m/s. The wedge can roll
  freely
• on the ground and is initially at rest. Knowing
  that the
• coefficient of restitution between the wedge and
  the
• sphere is 0.50 and that the inclined surface of
  the
• wedge forms an angle of 400 with the horizontal,
• determine (a) the velocities of the sphere and
  wedge
• immediately after impact, (b) the energy lost due
  to
• impact. For coefficient of restitution we use the
  T axis
• e - (vAT2 vBT2)/(vAT1 vBT1)
• 0.5 - (vAT2 vBT2)/(-4 0)
• Since there is an unknown impulse in the y
  direction
• Use x direction for conservation of momentum
• S(mv)X1 S(mv)X2
• 2(4cos50) 6(0) 2(-vAX2) 6vBX2
• -vAX2 vAT2cos50 and vBT2 vBX2cos50
• 4 equations 4 unknowns vAX2 , vAT2 , vBT2 , and
  vBX2
• vBX2 1.13 m/s vAT2 1.27 m/s

N
T
y
x
55

• Problem 13.188 A 2 kg sphere A strikes the
  frictionless
• inclined surface of a 6 kg wedge B at a 900 angle
  with a
• velocity of magnitude 4 m/s. The wedge can roll
  freely
• on the ground and is initially at rest. Knowing
  that the
• coefficient of restitution between the wedge and
  the
• sphere is 0.50 and that the inclined surface of
  the
• wedge forms an angle of 400 with the horizontal,
• determine (a) the velocities of the sphere and
  wedge
• immediately after impact, (b) the energy lost due
  to
• Since there is an unknown impulse in the y
  direction
• Use x direction for conservation of momentum
• S(mv)X1 S(mv)X2
• 2(4cos50) 6(0) 2(-vAX2) 6vBX2
• -vAX2 vAT2cos50 and vBT2 vBX2cos50
• 4 equations 4 unknowns vAX2 , vAT2 , vBT2 , and
  vBX2
• vBX2 1.13 m/s vAT2 1.27 m/s
• Before impact the kinetic energy is (1/2)2(-4)2
  16 Nm

N
T
y
x
56

• Problem 13.188 A 2 kg sphere A strikes the
  frictionless
• inclined surface of a 6 kg wedge B at a 900 angle
  with a
• velocity of magnitude 4 m/s. The wedge can roll
  freely
• on the ground and is initially at rest. Knowing
  that the
• coefficient of restitution between the wedge and
  the
• sphere is 0.50 and that the inclined surface of
  the
• wedge forms an angle of 400 with the horizontal,
• determine (a) the velocities of the sphere and
  wedge
• immediately after impact, (b) the energy lost due
  to
• Since there is an unknown impulse in the y
  direction
• Use x direction for conservation of momentum
• S(mv)X1 S(mv)X2
• 2(4cos50) 6(0) 2(-vAX2) 6vBX2
• -vAX2 vAT2cos50 and vBT2 vBX2cos50
• 4 equations 4 unknowns vAX2 , vAT2 , vBT2 , and
  vBX2
• vBX2 1.13 m/s vAT2 1.27 m/s
• Before impact the kinetic energy is (1/2)2(-4)2
  16 Nm
• After impact (1/2)2(1.27)2 (1/2)6(1.13)2 5.44
  Nm
• Energy Loss is 16 5.44 10.56 Nm

N
T
y
x