Kinematics and force example

A 1900 kilogram truck has an initial speed of 12 meters per second. The driver applies the brakes, and the truck stops in 3.1 seconds. What is the best estimate of the magnitude of the average braking force on the truck? Pause this video, see if you can work this out.

All right, probably the simplest way to approach this is to say, well, we could figure out the magnitude of the acceleration, the average acceleration on that truck as it comes to a stop. Then we could use Newton's second law, F = ma, because we know its mass. If we know the mass and the magnitude of the acceleration, we could use Newton's second law to come up with the magnitude of the average braking force. They just want us to estimate it, and we can see that these choices are pretty far apart. So, an estimate will serve us well.

So first, let's think about the acceleration. The acceleration is going to be our change in velocity over change in time. That is just going to be our final velocity minus our initial velocity in the numerator. So our final velocity is 0 meters per second; we come to a stop, minus our initial velocity, that's 12 meters per second. The convention that we'll assume, and it's typical, is that if we're moving to the right, it's positive; if we're moving to the left, it's negative. It's moving to the right at 12 meters per second, so we're subtracting that positive velocity out.

This is our change in velocity over our change in time. Well, 3.1 seconds elapses. So over 3.1 seconds, what is this going to be approximately equal to? Well, let's see. Negative 12 over 3.1 is going to be approximately 4. Once again, I am estimating; it might be a little bit closer to, oh, and it's going to be approximately equal to negative 4. It might be a little bit closer to negative 3.9, around there, but I'll go with negative 4. The units are meters per second squared.

Now we use Newton's second law to think about the magnitude of the braking force. This makes sense that the acceleration is negative, that our velocity is in this direction, but our acceleration is in the other direction. It is slowing down; we're getting lower and lower velocities. Our force is going to be in the same direction; our net force is in the same direction as that acceleration. It's going to be to the left, so if we had a sign on it, it would be negative, but we just care about the magnitude.

When we think about Newton's second law, we'll also just look at the magnitude of the acceleration. So, Newton's second law tells us the magnitude of the force needs to be equal to the mass times the magnitude of the acceleration. This is going to be equal to—let me write an approximation here because I approximated this. It's going to be approximately 1900 kilograms times 4 meters per second squared.

I didn't feel the need to write the negative there because I just want to get the magnitude; I care about the absolute value right now. If I do this, this is going to be approximately, what, 7600 kilograms? Now, if I look at the choices, I don't see 7600, but the closest one over here is 7400. I feel good about that because the real value here might be closer to negative 3.9 meters per second squared. If this was 3.9 right over here, 3.9 times 1900, well, that gets us a lot closer to this right here. So, I like this choice, and all the other ones are way off from our estimate.