Comparing instantaneous velocity and average velocity
A ball is dropped from a height of 64 feet. Its height above ground (in feet)
seconds later is given by
What is the instantaneous velocity of the ball when it hits the ground?
What is the average velocity during its fall?
The first thing to do is determine how long it takes the ball to reach the ground. To do this, set
Solving
we get
so it take 2 seconds for the ball to reach the ground.
The instantaneous velocity of the ball as it strikes the ground is
Since
we obtain
The average velocity of the ball during its fall is
A particle moves along a coordinate axis in the positive direction to the right. Its position at time
is given by
Find
and
and use these values to answer the following questions.
Is the particle moving from left to right or from right to left at time
Is the particle speeding up or slowing down at time
Begin by finding
and
and
Evaluating these functions at
we obtain
and
Because
the particle is moving from right to left.
Because
and
velocity and acceleration are acting in opposite directions. In other words, the particle is being accelerated in the direction opposite the direction in which it is traveling, causing
to decrease. The particle is slowing down.
The position of a particle moving along a coordinate axis is given by
Find
At what time(s) is the particle at rest?
On what time intervals is the particle moving from left to right? From right to left?
Use the information obtained to sketch the path of the particle along a coordinate axis.
The velocity is the derivative of the position function:
The particle is at rest when
so set
Factoring the left-hand side of the equation produces
Solving, we find that the particle is at rest at
and
The particle is moving from left to right when
and from right to left when
[link] gives the analysis of the sign of
for
but it does not represent the axis along which the particle is moving.
Since
on
the particle is moving from left to right on these intervals.
Since
on
the particle is moving from right to left on this interval.
Before we can sketch the graph of the particle, we need to know its position at the time it starts moving
and at the times that it changes direction
We have
and
This means that the particle begins on the coordinate axis at 4 and changes direction at 0 and 20 on the coordinate axis. The path of the particle is shown on a coordinate axis in
[link] .