7. The Physics Behind a Police Chase: How Fast Must They Drive?

If you've ever wondered about the mathematics and physics behind a police chase, you're not alone. There are many variables to consider, such as the speeds of the vehicles involved, the time delay before the police begin pursuit, and how quickly the police aim to catch up. To make this concept easier to understand, we'll use two hypothetical scenarios involving a car driving at 100 kilometers per hour (kph) and a police car aiming to catch up.

In our first scenario, imagine you're driving at 100 kph. There's a delay of 5 seconds before the police notice and start to pursue you. The question then becomes, "If the police want to reach you in 30 seconds after this 5-second delay, how fast do they need to drive to catch up?"

To answer this, we'll first need to convert your speed from kph to meters per second (m/s) for the sake of convenience. 100 kph is equivalent to approximately 27.78 m/s. During the initial 5 seconds, while you continue to drive and before the police begin their pursuit, you will have traveled:

Distance = Speed x Time = 27.78 m/s x 5 s = 138.9 meters.

In the next 30 seconds, you would cover an additional distance at the same speed:

Distance = Speed x Time = 27.78 m/s x 30 s = 833.4 meters.

The total distance the police need to cover in 30 seconds is the sum of these two distances, which equals 972.3 meters.

To find out how fast the police need to drive to cover this distance in 30 seconds, we use the formula Speed = Distance / Time:

Speed = 972.3 m / 30 s = 32.41 m/s.

To convert this back to kilometers per hour, we multiply by 3.6 (since 1 m/s = 3.6 km/hr):

Speed = 32.41 m/s x 3.6 km/hr = 116.68 kph.

So, in this scenario, the police would need to drive at a speed of approximately 117 kph to catch up to you in 30 seconds after a 5-second delay.

Our second scenario is a twist on the first. What if the police are able to drive at a higher speed, say 120 kph (or 33.33 m/s)? How long will it take them to reach you?

With a head start of 138.9 meters (the distance covered in the initial 5 seconds), we'll need to find out the time it takes for the police to close this gap. To do this, we use the relative speed, which is the difference between the police's speed and your speed:

Relative speed = Speed of police - Your speed = 33.33 m/s - 27.78 m/s = 5.56 m/s.

Finally, we use this relative speed to calculate the time it takes for the police to catch up:

Time = Distance / Speed = 138.9 m / 5.56 m/s = 25 seconds.

So, after the initial 5 seconds delay, it will take approximately 25 additional seconds for the police to reach you, making for a total of 30 seconds from when you pass the police.

These calculations serve as simplified models of a police pursuit, assuming perfect conditions such as instant acceleration and clear roads. Real-world factors such as acceleration time, road conditions, traffic laws, and vehicle limitations can significantly alter these numbers. Moreover, it's important to remember that exceeding the speed limit is both illegal and unsafe. Understanding the science behind these scenarios can be fascinating, but it should not encourage dangerous driving. Stay safe and respect speed limits!

So if the police say "yeah, we needed to drive this fast to reach you", then you can show them some very basic math and logic...

Please note that in these calculations it is assumed that police will instantly accelerate to for example 120kph, meaning that 120kph is the average speed of the police car, because they show you average of the speed they were travelling over certain distance. So they were slower at start when their speed ramped up, but they vere driving faster than that 120kph so they got 120kph average speed.