Final answer:
To find the time Michael will catch Robert, a physics problem involving kinematics is solved using the initial velocity, acceleration, and distance between the runners. Michael's position over time is described by a kinematic equation, considering his initial deficit and acceleration, while Robert moves at a constant speed.
Step-by-step explanation:
In this physics problem, we are asked to determine how long it will take Michael to catch up to Robert in a long-distance race. Michael starts 75 meters behind Robert. Robert runs at a constant velocity, but Michael accelerates at a constant rate, which will eventually allow him to catch up. To solve this, we use the kinematic equations for uniformly accelerated motion.
Let's denote the distance Michael needs to cover to catch Robert as d, Michael's initial velocity as vi (3.8 m/s), his acceleration as a (0.15 m/s2), and the time it takes to catch Robert as t. We'll also denote Robert's velocity as vr (4.2 m/s).
As Robert is running at a constant velocity, the distance he covers in time t is:
dr = vr × t
As Michael is accelerating, the distance he covers is governed by the following equation:
dm = vi × t + 0.5 × a × t2
Michael needs to cover his initial 75-meter deficit plus the distance that Robert runs in the same time, therefore:
75 + dr = dm
75 + vr × t = vi × t + 0.5 × a × t2
Now we just insert the values (75 + 4.2 × t = 3.8 × t + 0.5 × 0.15 × t2) and solve the quadratic equation for t to find the time it will take Michael to catch Robert.