Question

A runner of mass 57.5 kg starts from rest and accelerates with a constant acceleration of \(1.25 \mathrm{~m} / \mathrm{s}^{2}\) until she reaches a velocity of \(6.3 \mathrm{~m} / \mathrm{s}\). She then continues running with this constant velocity. a) How far has she run after 59.7 s? b) What is the velocity of the runner at this point?

Short answer

Answer: The runner covers a total distance of 360.3 meters and her velocity at 59.7 seconds is 6.3 m/s.

Step-by-step solution

a) Finding the distance run after 59.7 seconds

Step 1: Finding the time taken to reach a constant velocity of 6.3 m/s We can use the equation \(v = u + at\) : Where \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time. Given, \(v = 6.3 m/s\), \(u = 0 m/s\), and \(a = 1.25 m/s^2\). Using these values, we can find \(t\): \(6.3 = 0 + 1.25t\) Solving for \(t\), we get: \(t = 5.04s\) Step 2: Finding the distance covered during acceleration We can use the equation \(s = ut + \frac{1}{2}at^2\) : Where \(s\) is the distance, \(u\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time. Given, \(u = 0 m/s\), \(a = 1.25 m/s^2\), and \(t = 5.04 s\). Using these values, we can find \(s\) during acceleration: \(s = 0(5.04) + \frac{1}{2}(1.25)(5.04)^2\) Solving for \(s\), we get: \(s = 16.0 m\) Step 3: Finding the distance covered while running at a constant speed The total time was 59.7 s, and it took the runner 5.04 s to reach a constant speed of 6.3 m/s. Therefore, the time running at constant speed is \(59.7 - 5.04 = 54.66 s\). Now we use the equation \(s = vt\), we get: \(s = (6.3)(54.66)\) Solving for \(s\), we get: \(s = 344.3 m\) Step 4: Finding the total distance covered The total distance covered is the sum of the distance during acceleration and the distance while running at a constant speed, thus: Total distance = \(16.0 m + 344.3 m = 360.3 m\)

b) Finding the velocity of the runner at 59.7 seconds

Since it took the runner 5.04 seconds to reach a constant.velocity of 6.3 m/s and the total.time of the.runner was 59.7 seconds, she has been.running at a constant 6.3 m/s.for we 54.66 seconds. As she is running at a constant speed, there is no additional.step needed: Velocity = 6.3 m/s