Worked example: Motion problems with derivatives | AP Calculus AB | Khan Academy

A particle moves along the x-axis. The function x of t gives the particle's position at any time t is greater than or equal to zero, and they give us x of t right over here. What is the particle's velocity v of t at t is equal to 2?

So, pause this video, see if you can figure that out. Well, the key thing to realize is that your velocity as a function of time is the derivative of position. So, this is going to be equal to— we just take the derivative with respect to t up here.

The derivative of t to the third with respect to t is 3t squared. If that's unfamiliar, I encourage you to review the power rule. The derivative of negative 4t squared with respect to t is negative 8t, and the derivative of 3t with respect to t is plus 3. The derivative of a constant doesn't change with respect to time, so that's just 0.

Now, we have velocity as a function of time. If we want to know our velocity at time t equals 2, we just substitute 2 wherever we see the t's. So it's going to be 3 times 4. 3 times 2 squared, so it's 12 minus 8 times 2, minus 16, plus 3, which is equal to negative one.

You might say negative one by itself doesn't sound like a velocity. Well, if they gave us units, if they told us that x was in meters and that t was in seconds, then x would be— well, I already said would be in meters, and velocity would be negative one meters per second.

You might also be saying, "Well, what does the negative mean?" Well, that means that we are moving to the left. Remember, we're moving along the x-axis. So, if our velocity is negative, that means that x is decreasing or we are moving to the left.

What is the particle's acceleration a of t at t equals 3? So pause this video again and see if you can do that. Well, here the realization is that acceleration as a function of time is just the derivative of velocity, which is the second derivative of our position. This is just going to be equal to the derivative of this right over here.

So I'm just going to get the derivative of 3t squared with respect to t is 6t. The derivative of negative 8t with respect to t is minus 8, and the derivative of constants is 0. So, it's just going to be 6t minus 8.

So, our acceleration at time t equals 3 is going to be 6 times 3, which is 18 minus 8, so minus 8, which is going to be equal to positive 10.

All right, now they ask us what is the direction of the particle's motion at t equals two. Well, I already talked about this, but pause this video and see if you can answer that yourself.

Well, we've already looked at the sign right over here. The fact that we have a negative sign on our velocity means we are moving towards the left. So I'll fill that in right over there.

At t equals 3, is the particle's speed increasing, decreasing, or neither? So pause this video and try to answer that.

All right, now we have to be very careful here. If it says, "Is the particle's velocity increasing, decreasing, or neither?" then we would just have to look at the acceleration. We see that the acceleration is positive, and so we know that the velocity is increasing.

But here, they're not saying velocity, they're saying speed. Just as a reminder, speed is the magnitude of velocity. For example, at time t equals two, our velocity is negative one. If the units were meters per second, it would be negative one meters per second, but our speed would just be one meter per second.

Speed, you're not talking about the direction, so you would not have that sign there. In order to figure out if the speed is increasing or decreasing or neither, if the acceleration is positive and the velocity is positive, that means the magnitude of your velocity is increasing, so that means your speed is increasing.

If your velocity is negative and your acceleration is also negative, that also means that your speed is increasing. But if your velocity and acceleration have different signs, that means that your speed is decreasing. The magnitude of your velocity would be becoming less.

So let's look at our velocity at time t equals three. Our velocity at time 3, we just go back right over here, it's going to be 3 times 9, which is 27, 3 times 3 squared minus 24 plus 3 plus 3.

So this is going to be equal to 6. Our velocity and acceleration are both— you could say in the same direction. They are both positive, and so our velocity is only going to become more positive, or the magnitude of our velocity is only going to increase.

So our speed is increasing. If our velocity was negative at time t equals 3, then our speed would be decreasing because our acceleration and velocity would be going in different directions.