Final answer:
The police car takes 10 seconds to reach its maximum speed of 20 m/s. By this time, the thief is 50 meters ahead, and since the police car cannot exceed this speed, it cannot catch up to the thief. Hence, the total time is 10 seconds, but the thief is not caught.
Step-by-step explanation:
Time Taken by Police Car to Catch Thief
To solve the mathematical problem of how long it takes for the police car to catch the thief, we need to consider two phases of the police car's motion: acceleration to its maximum velocity, and then maintaining that maximum velocity.
Firstly, we calculate the time it takes for the police car to reach its maximum velocity of 20 m/s from rest with an acceleration of 2 m/s². Using the equation v = u + at, where v is the final velocity, u is the initial velocity (0 m/s), and a is the acceleration, we solve for t.
t = (v - u) / a
t = (20 m/s) / (2 m/s²) = 10 s
The distance covered by the police car during this acceleration phase can be found using s = ut + (1/2)at². Substituting the values, we get:
d = (0 m/s)(10 s) + (1/2)(2 m/s²)(10 s) ²
d = 100 m
By this time, the thief has traveled d_t = v_t × t where v_t is the thief's velocity (15 m/s) and t is the time (10 s).
d_t = 15 m/s × 10 s = 150 m
So, by the time the police car reaches 20 m/s, the thief is 50 m ahead. As the police car can now travel at the same speed as the thief (20 m/s), it will no longer close the gap. Therefore, the total time taken by the police car when the thief is caught is 10 seconds (the time it took to reach maximum speed), and the thief is not actually caught.