The chart shows the time, initial velocity, and final velocity of three riders.
| Rider | Time | Initial velocity | Final velocity |
| :---------- | :----- | :--------------- | :------------- |
| Gabriella | 10 sec | 55 | 32 |
| Franklin | 8.5 sec | 50 | 50 |
| Kendall | 6 sec | 53.2 | 67 |
Which best describes the riders' final velocities?
A. Gabriella is speeding up at the same rate that Kendall is slowing down.
B. Gabriella is slowing down at the same rate that Kendall is speeding up.
C. Gabriella and Franklin are both slowing down, and Kendall is accelerating.
D. Gabriella is slowing down, and Kendall and Franklin are accelerating.
Answer (1)
Calculate Gabriella's acceleration: a G = 10 32 − 55 = − 2.3 m/s 2 , indicating she is slowing down.
Calculate Franklin's acceleration: a F = 8.5 50 − 50 = 0 m/s 2 , indicating constant velocity.
Calculate Kendall's acceleration: a K = 6 67 − 53.2 = 2.3 m/s 2 , indicating she is speeding up.
Conclude that Gabriella is slowing down at the same rate that Kendall is speeding up, and Franklin maintains a constant velocity. The answer is: Gabriella is slowing down at the same rate that Kendall is speeding up.
Explanation
Understanding the Problem We are given the initial and final velocities of three riders, Gabriella, Franklin, and Kendall, along with the time it took for them to reach their final velocities. We need to determine which statement best describes their motion (speeding up, slowing down, or constant velocity).
Calculating Acceleration To determine whether each rider is speeding up, slowing down, or maintaining a constant velocity, we need to calculate their acceleration. The formula for acceleration is: a = time final velocity − initial velocity We will calculate the acceleration for each rider.
Gabriella's Acceleration For Gabriella: Initial velocity = 55 Final velocity = 32 Time = 10 sec a G = 10 32 − 55 = 10 − 23 = − 2.3 m/s 2 A negative acceleration indicates that Gabriella is slowing down.
Franklin's Acceleration For Franklin: Initial velocity = 50 Final velocity = 50 Time = 8.5 sec a F = 8.5 50 − 50 = 8.5 0 = 0 m/s 2 Zero acceleration indicates that Franklin is maintaining a constant velocity.
Kendall's Acceleration For Kendall: Initial velocity = 53.2 Final velocity = 67 Time = 6 sec a K = 6 67 − 53.2 = 6 13.8 = 2.3 m/s 2 A positive acceleration indicates that Kendall is speeding up.
Analyzing the Statements Now, let's analyze the statements:
Gabriella is speeding up at the same rate that Kendall is slowing down: This is incorrect because Gabriella is slowing down, and Kendall is speeding up.
Gabriella is slowing down at the same rate that Kendall is speeding up: This is correct because Gabriella's acceleration is -2.3 m/s 2 , and Kendall's acceleration is 2.3 m/s 2 . The magnitudes of their accelerations are the same, but the signs are opposite.
Gabriella and Franklin are both slowing down, and Kendall is accelerating: This is incorrect because Franklin has zero acceleration (constant velocity).
Gabriella is slowing down, and Kendall and Franklin are accelerating: This is incorrect because Franklin has zero acceleration (constant velocity).
Conclusion The best description of the riders' final velocities is: Gabriella is slowing down at the same rate that Kendall is speeding up.
Examples
Understanding acceleration is crucial in many real-world scenarios, such as designing vehicles, analyzing sports performance, and predicting the motion of objects. For example, when designing a car, engineers need to consider the acceleration and deceleration rates to ensure safety and efficiency. Similarly, in sports, athletes aim to maximize their acceleration to achieve better performance. By understanding these concepts, we can better analyze and predict motion in various contexts.
