Question

Shown are graphs of the position functions of two runners, A and B, who run a \(100\)meter race and finish in a tie.

(a) Describe and compare how the runners run the race.

(b) At what time is the distance between the runners the greatest?

(c) At what time do they have the same velocity?

Short answer

a. Runner A runs the entire race at constant velocity. Runner B starts the race slowly with respect to runner A but he finishes the race with a faster velocity than A.

b. The distance between the runners the greatest at the time between \(9\) and \(10\) seconds.

c. At almost \(9.5\) seconds they have the same velocity.

Step-by-step solution

(a) Step 1: Compare the runs of runner A and runner B

From the velocity-timegraph of the two runner, it is seen that runner A runs the race at constant velocity because slope of his velocity time graph is constant at every time \(t\).

On the other hand the runner B starts the race with a slower velocity than A because the slope of his running graph is lesser than runner A at the beginning. But he finishes the race with a faster velocity than A as the slope of his graph his greater than that of A.

(b) Step 2: Time when the distance between the two runners is greatest

The greatest distance between the runners be the largest perpendicular distance between the graph for runner A and runner B. From the graph we can estimate that the distance be greatest within the time \(9\) and \(10\) seconds.

(c) Step 3: The time when they have same velocity

The runner A and runner B will have same velocity at that time when both the graph have the same slope at a certain point. From the graph we can notice that at approximately \(9.7\) seconds the slopes of the two curves is the same. Hence approximately at \(9.7\) seconds they have the same velocity.