
A particle moves according to a law of motion s = f(t), t ≥ 0, where t is measured in seconds and s in feet.
(a) Find the velocity at time t.
(b) What is the velocity after 1 second?
(c) When is the particle at rest?
(d) When is the particle moving in the positive direction?
(e) Find the total distance traveled during the first 6 seconds.
(f) Draw a diagram like Figure 2 to illustrate the motion of the particle.
(g) Find the acceleration at time t and after 1 second.
(h) Graph the position, velocity, and acceleration functions for 0 ≤ t ≤ 6.
(i) When is the particle speeding up? When is it slowing down?
FIGURE 2
f(t) = t3 – 8t2 + 24t
(a)

To find: The velocity at time t.
Answer to Problem 1E
The velocity at time is
Explanation of Solution
Given:
The given equation is as below.
Calculation:
Calculate the velocity at time
Differentiate the equation (1) with respect to time.
Therefore, the velocity at time
(b)

To find: The velocity after 1 second.
Answer to Problem 1E
The velocity after 1 second is
Explanation of Solution
Calculate the velocity after 1 second.
Substitute 1 for
Therefore, the velocity after 1 second is
(c)

To find: The time when particle at rest.
Answer to Problem 1E
The particle never is at rest.
Explanation of Solution
Calculate the time when particle will be at rest.
The velocity will be zero, when the particle is at rest.
Substitute 0 for
From the above equation, the value of time t doesn’t exist. Therefore, the particle never is at rest.
(d)

To find: The particle moving in the positive direction.
Answer to Problem 1E
The velocity of particle always moves in positive direction.
Explanation of Solution
Calculate the time at which the particle will be moving in the positive direction.
If speed is positive, the particle moves in positive direction whereas the speed is negative, the particle moves in negative direction.
Substitute 0 for
Therefore, the velocity at
(e)

To find: The total distance traveled during the first 6 seconds.
Answer to Problem 1E
The total distance travelled during first 6 seconds is
Explanation of Solution
Calculate the total distance traveled during first 6 seconds.
Substitute 0 for
Substitute 6 for
Therefore, the total distance travelled during first 6 seconds is
(f)

To find: The diagram to illustrate the motion of the particle.
Answer to Problem 1E
The diagram is shown in the figure (1).
Explanation of Solution
Show the diagram to illustrate the motion of the particle as shown below in figure (1).
(g)

To find: The acceleration at time t and after 1 second.
Answer to Problem 1E
The acceleration at time is
Explanation of Solution
Calculate the acceleration at time t.
Differentiate the equation (2) with respect to t.
Therefore, the acceleration at time is
Calculate the acceleration after 1 second.
Substitute 1 for
Therefore, the acceleration after 1 second is
(h)

To sketch: The graph the position, velocity, and acceleration function for
Explanation of Solution
Calculate the position using the formula.
Substitute 0 for
Similarly, calculate the remaining values.
Calculate the value of
0 | 0 |
0.5 | 10.125 |
1 | 17 |
1.5 | 21.375 |
2 | 24 |
2.5 | 25.625 |
3 | 27 |
3.5 | 28.875 |
4 | 32 |
4.5 | 37.125 |
5 | 45 |
5.5 | 56.375 |
6 | 72 |
Calculate the velocity using the expression.
Substitute 0 for
Similarly, calculate the remaining values.
Calculate the value of
0 | 24 |
0.5 | 16.75 |
1 | 11 |
1.5 | 6.75 |
2 | 4 |
2.5 | 2.75 |
3 | 3 |
3.5 | 4.75 |
4 | 8 |
4.5 | 12.75 |
5 | 19 |
5.5 | 26.75 |
6 | 36 |
Calculate the acceleration using the formula.
Substitute 0 for
Similarly, calculate the remaining values.
Calculate the value of
0 | -16 |
0.5 | Z-13 |
1 | -10 |
1.5 | -7 |
2 | -4 |
2.5 | -1 |
3 | 2 |
3.5 | 5 |
4 | 8 |
4.5 | 11 |
5 | 14 |
5.5 | 17 |
6 | 20 |
Draw the position as a function of time curve as shown in the figure (1).
Draw the speed as a function of time curve as shown in the figure (2).
Draw the acceleration as a function of time curve as shown in the figure (3).
(i)

To find: The time when the particle is speeding up and slowing down.
Answer to Problem 1E
The time when the particle is speeding up is when time t is greater than
Explanation of Solution
Calculate the time when particle is speeding up and slowing down.
Substitute 0 for
Substitute
Substitute
Therefore, the acceleration is positive when the value of time is
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