Solve for time in constant velocity question (easy)

Question

A speeder drives by a cop car moving 50m/s. The cop takes 12 seconds to get things in order and then chases after the speeder, traveling 60 m/s. How much time will it take the cop to catch up to the speeder?

This is what I have so far:

$v_c= 60$ m/s

$x_c = 60(t-12)$

$v_s = 50$ m/s

$x_s = 50t$

I set the two equations equal to each other and I have

$$50t = 60(t-12)$$

$$t=72$$

However the solution says

$$60t = 600+50t$$

$$t=60$$

Why am I incorrect in my thinking? And shouldn't both methods of solving give the same answer?

Answer 1

You need to focus on what's been asked in the question.

It's asked How much time will it take for the cop to catch up to the speeder?, not after what time they meet or cross each other?.

As it's given that the cop takes $12$ seconds to get things in order and he starts after that, we need to find the time after the 12$^{th}$ second. So you have time = $t_{\text{used by you }}-12 = 72-12 = 60$.

The time $t$ you've considered gives the total time after which they meet and you just need to subtract the $12$ seconds the cop took to get things in order in order to find the time taken by the cop to catch the speeder.

Answer 2

The solution of $t=60$ is correct. The speeder is $600$m ahead of the cop when he starts at $60$ meters per second. His speed relative to the speeder is $10$ meters per second so it takes $60$ seconds to catch the speeder.