Concept explainers
(a) Use the data in Problem 3 to construct a smooth graph of position versus time. (b) By constructing tangents to the x(t) curve, find the instantaneous velocity of the car at several instants. (c) Plot the instantaneous velocity versus time and, from this information, determine the average acceleration of the car. (d) What was the initial velocity of the car?
(a)

The graph of position versus time.
Answer to Problem 10P
Therefore, the smooth graph of position versus time is shown in Figure I.
Explanation of Solution
The following table contains the data of position of the car at various time instants.
Draw the graph of position versus time for the derby car.
Figure I
In the shown graph, the position of the car at various time instants is plotted in the vertical axis against the time along horizontal direction.
Conclusion:
Therefore, the smooth graph of position versus time is shown in Figure I.
(b)

The instantaneous velocity of the car at various time instants.
Answer to Problem 10P
The instantaneous velocity of the car at
Explanation of Solution
The following table contains the data of position of the car at various time instants.
The instantaneous velocity is the slope of the tangent of the position versus time graph at an instant.
Formula to calculate the slope of the tangent is,
Here,
Draw the tangent line at the time instant of
Figure II
Substitute
Therefore, the instantaneous velocity of the car at
Draw the tangent line at the time instant of
Figure III
Substitute
Therefore, the instantaneous velocity of the car at
Draw the tangent line at the time instant of
Figure IV
Substitute
Therefore, the instantaneous velocity of the car at
Draw the tangent line at the time instant of
Figure V
Substitute
Therefore, the instantaneous velocity of the car at
Draw the tangent line at the time instant of
Figure VI
Substitute
Conclusion:
Therefore, the instantaneous velocity of the car at
(c)

The average acceleration of the car.
Answer to Problem 10P
The average acceleration of the car is
Explanation of Solution
The following table contains the instantaneous velocity of the car at various times instant.
The graph of instantaneous velocity versus time for the derby car is shown below.
Figure VII
The figure VII shows that velocity of the car increases linearly, it means the acceleration of the car is constant throughout the motion.
Thus, the slope of this graph gives the average acceleration of the car.
Formula to calculate the slope of versus time graph is,
Here,
Substitute
From the graph, the slope of the graph is
Conclusion:
Therefore, the average acceleration of the car is
(d)

The initial velocity of the car.
Answer to Problem 10P
The initial velocity of the car is zero.
Explanation of Solution
The equation for the velocity of the car obtained from the graph is,
The first equation of motion gives the velocity of an object at any instant.
Here,
Compare equation (I) and (II).
Thus, the initial velocity of the car is zero.
Conclusion:
Therefore, the initial velocity of the car is zero.
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