Calculus

Applications of the Derivative

Applications of Derivative Logo

Rectilinear Motion

Rectilinear motion is a motion of a particle or object along a straight line.

Position is the location of object and is given as a function of time s (t) or x (t).

Velocity is the derivative of position:

v=dxdt.

Acceleration is the derivative of velocity:

a=dvdt.

The position and velocity are related by the Fundamental Theorem of Calculus:

t1t2v(t)dt=x(t)|t1t2=x(t2)x(t1),

where t1tt2. The quantity x(t2)x(t1) is called a displacement. The displacement is represented by the area under the graph of the velocity v(t).

Displacement is the area under the velocity graph.
Figure 1.

Similarly, since acceleration is the rate at which the velocity changes, we have

where the quantity is the net change in velocity in the time interval

Speed is the absolute value of velocity, i.e. speed is always positive.

The average speed is defined as

The total distance a particle travels between time and time is given by

Solved Problems

Example 1.

The graph in Figure below shows a particle's velocity moving along a coordinate line. At the position is

  1. Sketch the acceleration vs. time graph corresponding to this velocity vs. time graph;
  2. Sketch the graph of position vs. time corresponding to the velocity vs. time graph;
  3. Determine the average speed of the particle between and

Solution.

Velocity vs. time graph
Figure 2.

1. Acceleration vs. time graph.

Acceleration vs. time graph
Figure 3.

2. Position vs. time graph.

Position vs. time graph
Figure 4.

3. The average speed of the particle between and

Consider two intervals:

Though the particle returns to the initial position at the total distance traveled from to is equal to

The average speed is equal to

Example 2.

A particle moving on a line is at position at time where is in seconds. At which time (if any) does the particle change its direction?

Solution.

Find the particle's velocity by differentiating position function:

As you can see, the velocity becomes equal to zero at and Hence, the particle changes its direction at the indicated times.

Example 3.

A particle moves along the axis according to the equation where time and is measured in seconds. Find the time when the particle's velocity and acceleration are equal.

Solution.

We differentiate the position function successively to determine velocity and acceleration:

By equating and we obtain

or

This equation has a positive root

Answer:

Example 4.

The position function of a particle moving along the axis is given by Find the open intervals when the particle is moving to the left.

Solution.

Find the particle's velocity by differentiating the position function:

Solve the quadratic equation:

Rewrite the velocity function in factored form:

We see that the velocity is negative when . In that time interval, the particle is moving to the left.

Example 5.

A particle moves along the axis so that its coordinate obeys the law where is in meters and is in seconds.

  1. Find the particle's velocity;
  2. Find the particle's acceleration;
  3. Determine the average speed of the particle between and

Solution.

1. Particle's velocity.

Take the derivative of

Hence, the velocity of the particle is given by the equation

2. Particle's acceleration.

To find acceleration, we differentiate the velocity function:

As you can see, the particle moves with a constant acceleration.

3. Particle's average speed.

First we determine the position of the particle at and

Compute the average speed in the given time interval:

Example 6.

An object moves along the axis according to the law where is in meters, is in seconds.

  1. Find time when the acceleration is zero;
  2. Calculate the object's velocity at this instant.

Solution.

The velocity is obtained by successive differentiation of with respect to time

Similarly, to get the acceleration, we differentiate the velocity with respect to time

Determine when the acceleration is equal to zero:

Calculate the object's velocity at

Example 7.

Find the integral expression that would result in the total distance traveled on the interval [0,3] if the velocity is given by

Solution.

Particle's velocity graph
Figure 5.

To find the total distance we need to integrate the speed function, i.e. the absolute value of the velocity. Note that the velocity changes sign at Therefore we split the interval into two intervals and The total distance traveled by the particle on the interval is expressed in the form:

Given that the velocity is negative on the first interval and positive on the second interval, we get

Rearranging the terms, we have

Example 8.

A particle is moving along the -axis so that its position at time is given by the equation Determine the acceleration of the particle when the velocity is zero.

Solution.

Find the particle's velocity by differentiating the position function:

Continue differentiating to find the acceleration:

The velocity is zero when time is

Substituting this time value, we find the acceleration at this instant:

Example 9.

When two particles start at the origin with velocities how many times in the interval will their speeds be equal?

Solution.

We solve this problem graphically. Let's draw the graphs of the speed functions and on the interval

Speed functions of two particles
Figure 6.

We see that the curves intersect times on the interval

Example 10.

A particle moves along a straight line according to the equation where is in meters, is in seconds. Find the total distance traveled by the particle after 6 seconds.

Solution.

To find the total distance traveled by a particle, we need to take the integral of of the speed

Determine the particle's velocity:

We see that the velocity is negative for and positive when Hence, we split up the integral into the following two components:

Given that the velocity is negative in the first integral and positive in the second, we obtain:

This yields:

So, the total distance traveled by the particle is equal to