Question

Pablo is running in a half marathon at a velocity of 3.0 m/s. Another runner, Jacob, is 53.0 meters behind Pablo with the same velocity. Jacob begins to accelerate at 0.044 m/s2. How long does it take Jacob to catch Pablo?

Answer 1

To determine how long it takes for Jacob to catch up with Pablo, we need to find the time it takes for Jacob to cover the initial distance between them (53.0 meters) and match Pablo's velocity. After that point, both runners will have the same velocity, and Jacob will catch up.

Let's break down the problem step by step:

1. Find the time it takes for Jacob to reach the velocity of 3.0 m/s: Jacob's initial velocity is 0 m/s, and he accelerates at a rate of 0.044 m/s^2. We can use the equation of motion: v = u + at, where: v = final velocity (3.0 m/s), u = initial velocity (0 m/s), a = acceleration (0.044 m/s^2), and t = time.

   Rearranging the equation, we get: t = (v - u) / a.

   Plugging in the values: t = (3.0 m/s - 0 m/s) / 0.044 m/s^2. t = 68.18 seconds (rounded to two decimal places).

   Therefore, it takes Jacob approximately 68.18 seconds to reach a velocity of 3.0 m/s.

2. Calculate the distance covered by Jacob during this time: Using the equation of motion: s = ut + (1/2)at^2, where: s = distance covered, u = initial velocity, a = acceleration, t = time.

   Plugging in the values: s = 0 m/s * 68.18 s + (1/2) * 0.044 m/s^2 * (68.18 s)^2. s ≈ 82.84 meters.

   Therefore, Jacob covers approximately 82.84 meters during the time it takes him to reach a velocity of 3.0 m/s.

3. Calculate the time it takes for Jacob to close the remaining distance (53.0 meters): The relative velocity between Jacob and Pablo will be 3.0 m/s since they will both be running at that speed. Therefore, the remaining distance of 53.0 meters will be closed at this relative velocity.

   t = s / v, where: t = time, s = remaining distance, and v = relative velocity.

   Plugging in the values: t = 53.0 m / 3.0 m/s. t ≈ 17.67 seconds (rounded to two decimal places).

   Therefore, it takes Jacob approximately 17.67 seconds to close the remaining distance of 53.0 meters.

4. Calculate the total time it takes for Jacob to catch Pablo: The total time is the sum of the time it took for Jacob to reach Pablo's velocity (68.18 seconds) and the time it took to close the remaining distance (17.67 seconds).

   Total time = 68.18 s + 17.67 s. Total time ≈ 85.85 seconds (rounded to two decimal places).

   Therefore, it takes Jacob approximately 85.85 seconds to catch up with Pablo.