Worked example: Motion problems with derivatives
- [Instructor] 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 two? 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. And so this is going to be equal to, we just take the derivative with respect to t up here. So derivative of t to the third with respect to t is three t squared. If that's unfamiliar, I encourage you to review the power rule. The derivative of negative four t squared with respect to t is negative eight t. And derivative of three t with respect to t is plus three. Derivative of a constant doesn't change with respect to time, so that's just zero. And so here we have velocity as a function of time. And so if we want to know our velocity at time t equals two, we just substitute two wherever we see the t's. So it's gonna be three times four, three times two squared, so it's 12 minus eight times two, minus 16, plus three, which is equal to negative one. And 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, well, 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 means? Well, that means that we are moving to the left. Remember, we're moving along the x-axis. So if our velocity's negative, that means that x is decreasing or we're moving to the left. What is the particle's acceleration a of t at t equals three? So pause this video again, and see if you can do that. Well, here the realization is that acceleration is a function of time. It's just the derivative of velocity, which is the second derivative of our position, which is just going to be equal to the derivative of this right over here. And so I'm just going to get derivative of three t squared with respect to t is six t. Derivative of negative eight t with respect to t is minus eight. And derivative of a constant is zero. So it's just going to be six t minus eight. So our acceleration at time t equals three is going to be six times three, which is 18, minus eight, so minus eight, 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 three, 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. And just as a reminder, speed is the magnitude of velocity. So, for example, at time t equals two, our velocity is negative one. If the units were meters and 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. And so 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, well, that means that your speed is decreasing. The magnitude of your velocity would become less. So let's look at our velocity at time t equals three. Our velocity at time three, we just go back right over here, it's going to be three times nine, which is 27, three times three squared, minus 24 plus three, plus three. So this is going to be equal to six. So our velocity and acceleration are both, you could say, in the same direction. They are both positive. And so our velocity's 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 three, then our speed would be decreasing because our acceleration and velocity would be going in different directions.