(a) Well, well, well, looks like we have a high-speed pursuit on our hands! To find out how much time elapses before the police catches up to the motorist, we need to first determine the distance the motorist travels before the police starts the pursuit.
Since the motorist is traveling at a constant speed of 15m/s and the speed limit is 10m/s, we can calculate the distance covered before the police starts as follows:
Distance covered = Speed × Time
Distance covered by motorist = 15m/s × Time
Now, let's look at the police officer. The officer starts from rest and undergoes constant acceleration of 3m/s^2. To catch up to the motorist, the equation we'll need is:
Distance covered = Initial velocity × Time + 0.5 × Acceleration × Time^2
But here's the thing, we want this distance to be equal to the distance covered by the motorist. So we can set up the equation:
Distance covered by police = Distance covered by motorist
(Initial velocity of police × Time + 0.5 × Acceleration of police × Time^2) = (15m/s × Time)
Now, all we need to do is solve for Time. Get your calculators ready!
(b) Once we've determined the time it takes for the police to catch up to the motorist, we can find the officer's speed at that point. To calculate the officer's speed, we'll use the equation:
Officer's speed = Initial velocity + Acceleration × Time
Plug in the values, do the math, and voila!
(c) Finally, to calculate the total distance each vehicle has traveled at that point, we need to add up the distance covered by each of them. For the motorist, it will be:
Distance covered by motorist = Speed of motorist × Time
And for the police officer, it will be:
Distance covered by police = Initial velocity × Time + 0.5 × Acceleration × Time^2
Once again, grab your calculators and let's crunch those numbers!
Remember, in the world of high-speed pursuits, calculations can be intense but comedy can lighten the mood. Stay safe out there, folks!