Position,%20Velocity%20and%20Acceleration

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Position, Velocity and Acceleration

• 3.2

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Position, Velocity and Acceleration

• All fall under rectilinear motion
• Motion along a straight line
• We are normally given a function relating the
  position of a moving object with respect to time.
• Velocity is the derivative of position
• Acceleration is the derivative of velocity

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Position, Velocity and Acceleration

• Position
• s(t) or x(t)
• Velocity
• v(t) or s(t)
• Acceleration
• a(t) or v(t) or s(t)
• Speed is the absolute value of velocity

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Example

• If the position of a particle at time t is given
  by the equation below, find the velocity and
  acceleration of the particle at time, t 5.

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Position, Velocity and Acceleration

• When velocity is negative, the particle is moving
  to the left or backwards
• When velocity is positive, the particle is moving
  to the right or forwards
• When velocity and acceleration have the same
  sign, the speed is increasing
• When velocity and acceleration have opposite
  signs, the speed is decreasing.
• When velocity 0 and acceleration does not, the
  particle is momentarily stopped and changing
  direction.

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Example

• If the position of a particle is given below,
  find the point at which the particle changes
  direction.

Changes direction when velocity 0 and
acceleration does not
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Example

• Using the previous function, find the interval of
  time during which the particle is slowing down.

V(t) 0 at 2 and 6, a(t) 0 at 4
4
6
0
2
Particle is slowing down when, 0 lt t lt 2 4 lt t lt
6
t
v(t) - -
a(t) - -
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Example
When velocity 0
When does this occur?

• How far does a particle travel between the eighth
  and tenth seconds if its position is given by

To find the total distance we must find if the
particle changes directions at any time in the
interval
The object may travel forward then backwards,
thus s(10) s(8) is really only the displacement
not the total distance!
3 is not in our interval so it will not affect
our problem!
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Example

• How far does a particle travel between zero and
  four seconds if its position is given by

Divide into intervals 0?2 and 2?4
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At any time t, the position of a particle moving
along an axis is
A. Find the bodys acceleration each time the
velocity is zero
C. Find the total distance traveled by the body
from t 0 to t 2
Velocity 0 at 1!
Divide into intervals 0?1 and 1?2
B. Find the bodys speed each time the
acceleration is zero
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At any time t, the position of a particle moving
along an axis is
A. When is the body moving forward? backwards?
1
3

v(t) -
Forward (0, 1) and (3, 8) Backwards from (1, 3)
B. When is the velocity increasing? decreasing?
0
1
2
3
Velocity increasing (1, 2) and (3, 8) Velocity
decreasing (0, 1) and (2, 3)
t
v(t) - -
a(t) - -