An exposure to Newtonian mechanics: part II

1
An exposure to Newtonian mechanics part II

• Solving 17 problems

Motivation Newtons concept of the Universe was
one of crystalline beauty. The future is
predictable. The past can be reconstructed.
The present can be completely
deconstructed. Today, we will explore some
examples.
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Our tools

• Key definitions
• v ?x/?t a ?v/?t vtotal v1 v2
• Newtons basics
• F ma F1 -F2 F12 Gm1m2/R122
• Expressions of energy
• K.E.½mv2 P.E.mgh
• Circular force Kinematics
• Fc mvc2/R x ½a ?t2 R(v2/g)sin2?

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Velocities

• 1) A juvenile delinquent skateboarding at
• 10 km/hr, throws a bottle forwards at
• 20 km/hr relative to him. How fast is the bottle
  travelling with respect to the Halloween pumpkin
  that it hits?
• Easy! Just use the velocity addition formula.

vtotal v1 v2 vtotal 10 km/hr 20 km/hr
30 km/hr 2) What if you were travelling at 2/3
the speed of light, and fired a probe at 2/3 the
speed of light. How fast does the star P-Umpkin
see the probe coming towards it? vtotal v1
v2 vtotal (2/3)c (2/3)c (4/3)c 4108 m/s.
NOTE THE USE OF c
4
Kinematics

• 3) A car drives 150 km in 4 hours. What is its
  speed?
• v ?x/?t 150 km / 4 hr 37.5 km/hr

4) A car accelerates from a stop to 100 km/hr in
6 seconds. What is its average acceleration? Firs
t, convert the change in velocity to m/sec 100
km/hr (1000 m/km) (1hr/3600 s) 27.8 m/s a
?v/?t 27.8 m/s / 6 s 4.6 m/s/s 4.6 m/s2
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Kinematics

• 5) Thelma and Louise plummet off the edge of
  the Grand Canyon. How long until they pancake?
• ?x ½at2 ? t22?x/g (2 1800 m)/(9.8 m/s2)
  367 s2
• t19 s
• How fast are they going at impact?
• vat gt (9.8 m/s2) 19 s
• v186 m/s 420 miles/hr

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Kinematics

• 6) At the strike of midnight, a new-years
  reveler shoots his .357 Remington, aimed upwards
  at a 45 angle. How far away does the bullet
  land?
• vmuzzle 46 m/s
• R(v2/g)sin2? (46 m/s)2/9.8 m/s (sin90)
• R216 m
• At what time does the bullet go through your
  living room window?
• (Magic vhorizontal vmuzzle cos? 32.5 m/s)
• t?x/v (216 m)/(32.5 m/s) 6.6 s
• T120007

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Newtons Laws

• 7) The space shuttles main engines and 2 SRBs
  provided a total thrust of about 31,000,000 N
  (force). The shuttle (unloaded) had a mass of
  about 2106 kg (2,000,000 kg).
• If an empty shuttle, with two SRBs, began
  travelling in space, what would be its
  acceleration?

Fma ? a F/m a 31,000,000 N / 2,000,000 kg
15.5 m/s2
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Newtons Laws
8) Your mass is 80 kg. How hard is the pull of
gravity, from an object as massive as the Earth,
from a distance of the Earths radius? F
GMEm/RE2 (6.6710-11 N m2/kg2) (5.971024
kg) (80 kg) (6,378,000
m)2 783 N. Since 4.45 N 1 pound, F176
pounds
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Keplers Third Law (1627)

• Planets orbiting the Sun follow the law that
• P2Da3
• Where P is the orbital period and a is the
  distance of the object from the Sun.
• 9) Let us look at this for the circular orbit
  case.

Is it that the orbits are simply larger, and take
longer to traverse? v ?x/?t, so ?t ?x/v ?x
C 2pa, ?t P 2pa/v, so P2 4p2a2/v2
(4p2/v2)a2 Da2 Close, but not quite Keplers
Third Law.
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Newton and Keplers Third Law

• P2Da3
• Where P is the orbital period and a is the
  distance of the object from the Sun.

10) Now let us try balancing the gravitational
force of attraction with the inward centripetal
force needed to maintain a circular orbit. F12
Gm1m2/R2 and Fc m1vc2/R, so Gm1m2/R2
m1vc2/R Gm/a v2 (I substituted a for R, as
in Keplers Third Law) Since P 2pa/v, therefore
v 2pa/P, and so Gm/a 4p2a2/P2, so P2
(4p2/Gm)a3 Keplers Third Law!
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Energy and free fall

• 11) How fast would something be moving if it
  accelerated at 1g, for a distance equal to the
  Earths diameter?
• Easy! Just balance P.E. and K.E.

mgh ½mv2 gD ½v2 v2 2gD, so v
(2gD)½ v (2gD)½ (29.8 m/s26.4106 m)½ v
11,200 m/s 25,000 mph! i.e, Mach 73 (the speed
of sound is about 343 m/s). The Space Shuttle
orbited at about 8000 m/s.
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Energy and free fall

• 12) How far would our object in 1g have to fall,
  to reach the speed of light?
• How does this distance compare to interplanetary
  and interstellar space?
• Easy! Just balance K.E. and P.E.

mgh ½mv2 gD ½c2 D c2/2g D (3108 m/s)2
/(2 9.8 m/s2 ) 4.6 1015 m Since 1 AU 1.5
1011 m, and 1 LY 9.46 1015 m D 3.1 104 AU
0.49 LY
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Length measure

• 13) Youre sitting at a railroad crossing, and
  the trainpassing at 50 km/hourtakes five
  minutes to go by. How long is the train?
• Easy! Just use the rate equation.
• (The length of the train is the same as just
  asking how far does the front of the train
  travel, in the time it took for the rear of the
  train to arrive.)

v ?x/?t ?x v?t ?x (50 km/hr 1 hr/60 min)
5 min 4.2 km 4200 m
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More gravity
14) Professional basketball players can typically
jump upwards about 1.10 m. How fast are they
jumping? Easy! Just balance K.E. and P.E.
mgh ½mv2 gh ½v2 v2 2gh, so v (2gh)½ (2
9.8 m/s21.1 m)½ v 4.6 m/s 10.4 mph 15) How
fast is an elevator falling if it plummets three
stories? (Estimate 1 story as 3.7 m 12 ft.) v
(2gh)½ (2 9.8 m/s2 11.1 m)½ v 14.7 m/s
32.9 mph
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Weight loss
16) How does your weight compare on a planet with
a different mass and radius? Make a ratio of
the law of gravity.
FE GMEm/RE2 FP GMPm/RP2, so (FP/ FE)
(GMPm/RP2)/(GMEm/RE2) (FP/ FE)
(MP/RP2)/(ME/RE2) (MP/ME) (RE/RP)2 Mars, M
0.107 ME, R 0.53 RE, FMars (0.107 ME/ME)
(RE/0.53 RE)2 0.38 FE
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Monkey in a tree
17) Consider a monkey in a tree. It knows you
intend to shoot it with your gun, which is
pointed directly at it. The monkey drops from the
tree the moment you fire the gun. What happens?
17
Not all physics is easy

• Rotations complicate things!
• Consider a spinning book!