Question

A relay race is run along a straight-line track of length \(300.0 \mathrm{m}\) running south to north. The first runner starts at the south end of the track and passes the baton to a teammate at the north end of the track. The second runner races back to the start line and passes the baton to a third runner who races \(100.0 \mathrm{m}\) northward to the finish line. The magnitudes of the average velocities of the first, second, and third runners during their parts of the race are \(7.30 \mathrm{m} / \mathrm{s}, 7.20 \mathrm{m} / \mathrm{s}\) and \(7.80 \mathrm{m} / \mathrm{s},\) respectively. What is the average velocity of the baton for the entire race? [Hint: You will need to find the time spent by each runner in completing her portion of the race.]

Short answer

Answer: The average velocity of the baton for the entire race is given by the formula: \(average\_velocity = \frac{100\,\mathrm{m}}{time_1 + time_2 + time_3}\) where time_1, time_2, and time_3 represent the time taken by each runner to complete their part of the race. To find the actual value of the average velocity, substitute the values of time_1, time_2, and time_3 from the solution and solve for the average velocity.

Step-by-step solution

Calculate the time taken by each runner

To find the time taken by each runner to complete their part of the race, we use the formula: time = distance / velocity. For the first runner: \(time_1 = \frac{300.0\,\mathrm{m}}{7.30\,\mathrm{m/s}}\) For the second runner: \(time_2 = \frac{300.0\,\mathrm{m}}{7.20\,\mathrm{m/s}}\) For the third runner: \(time_3 = \frac{100.0\,\mathrm{m}}{7.80\,\mathrm{m/s}}\)

Calculate the total time taken by all runners

Now we add up the times calculated in step 1 to find the total time taken by all three runners: \(total\_time = time_1 + time_2 + time_3\)

Calculate the total distance traveled by the baton

The first runner travels 300m northward, the second runner travels 300m southward, and the third runner travels 100m northward. Since the directions are opposite for the first two runners, their distances subtract. \(total\_distance = 300\,\mathrm{m} - 300\,\mathrm{m} + 100\,\mathrm{m} = 100\,\mathrm{m}\)

Calculate the average velocity of the baton for the entire race

To find the average velocity of the baton for the entire race, we divide the total distance by the total time: \(average\_velocity = \frac{total\_distance}{total\_time}\) Plug the values from steps 2 and 3 into the formula to get the final answer: \(average\_velocity = \frac{100\,\mathrm{m}}{time_1 + time_2 + time_3}\)