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Page 60: Practice Problems

Exercise 37

Solution 1

Solution 2

Step 1

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$textbf{(a)}$ Both cars have a positive slope, which means that their velocities are positive. The slope corresponding to the velocity of car B is greater than the one corresponding to car A, and $textbf{the steeper the slope, the greater the velocity.}$ Therefore, the velocity of car A is less than the velocity of car B.

Step 2

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$textbf{(b)}$ Using the time-position graph, $textbf{the initial position is determined by the intercept on the vertical axis.}$ In other words, it’s the value of position $x$ when $t=0$.

At $t=0$, car B is at $x_{B}=5 text{m}$, and car A is at $x_{A}=12.5 text{m}$, which means that the initial position of car A is greater than the initial position of car B.

Step 3

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$textbf{(c)}$ $textbf{To find the position of an object at a given time, find that value on the $x$ axis and trace from that point upward until you reach the straight line. From that point, trace sideways until you reach the $y$ axis.}$

Starting at $t=1 text{s}$ and moving up, we reach the line representing the velocity of car B. Moving left, we see that the value on the $y$ axis is exactly 10. Repeating the same procedure for car A, we find that the value on the $y$ axis is greater than 10. Thus, in the time period from $t=0$ to $t=1 text{s}$, car A is ahead of car B.

Result

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$textbf{(a)}$ The velocity of car A is less than the velocity of car B.

$textbf{(b)}$ The initial position of car A is greater than that of car B.

$textbf{(c)}$ From $t=0$ to $t=1 text{s}$, car A is ahead of car B.

Step 1

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a.) $textbf{Concept:}$
The velocity of car equals the slope of the graph and the y-intercept of the graph is the initial position of the car. By making use of these facts we will answer questions about the motions of the two cars.

$$
textbf{solution:}
$$

For car A the slope is smaller than the slope of the graph for car B. Therefore, we conclude that the velocity of car A $text{color{#4257b2} boxed{bf is less than} the velocity of car B.}$

Step 2

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b.) $textbf{Concept:}$
The velocity of car equals the slope of the graph and the y-intercept of the graph is the initial position of the car. By making use of these facts
we will answer questions about the motions of the two cars.

$$
textbf{solution:}
$$

For car A, the intercept on y-axis the graph is greater than the y-intercept of the graph for car B. Therefore, we found that the initial position of car A is $text{color{#4257b2} boxed{bf greater than} the initial position of car B.}$

Step 3

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c.) $textbf{Concept:}$
The velocity of car equals the slope of the graph and the y-intercept of the graph is the initial position of the car. By making use of these facts we will answer questions about the motions of the two cars.

$$
textbf{solution:}
$$

Car A has position that is larger than the position of car B during the time interval $t = 0$ to $t = 1 s$. Therefore, we found that car A $text{color{#4257b2} boxed{bf is ahead} of car B during that time interval.}$

Result

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