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:
Acceleration is the derivative of velocity:
The position and velocity are related by the Fundamental Theorem of Calculus:
where
Similarly, since acceleration is the rate at which the velocity changes, we have
where the quantity
Speed
The average speed
The total distance
Solved Problems
Example 1.
The graph in Figure
- Sketch the acceleration
vs. time graph corresponding to this velocity vs. time graph; - Sketch the graph of position
vs. time corresponding to the velocity vs. time graph; - Determine the average speed of the particle between
and
Solution.
1. Acceleration vs. time graph.
2. Position vs. time graph.
3. The average speed of the particle between and
Consider two intervals:
Though the particle returns to the initial position
The average speed is equal to
Example 2.
A particle moving on a line is at position
Solution.
Find the particle's velocity by differentiating position function:
As you can see, the velocity becomes equal to zero at
Example 3.
A particle moves along the
Solution.
We differentiate the position function successively to determine velocity and acceleration:
By equating
or
This equation has a positive root
Answer:
Example 4.
The position function of a particle moving along the
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
Example 5.
A particle moves along the
- Find the particle's velocity;
- Find the particle's acceleration;
- 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
Compute the average speed in the given time interval:
Example 6.
An object moves along the
- Find time
when the acceleration is zero; - Calculate the object's velocity at this instant.
Solution.
The velocity is obtained by successive differentiation of
Similarly, to get the acceleration, we differentiate the velocity
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.
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
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
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
Solution.
We solve this problem graphically. Let's draw the graphs of the speed functions
We see that the curves intersect
Example 10.
A particle moves along a straight line according to the equation
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
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