A car is stopped at a traffic light. It then travels along a straight road such that its distance from the light is given by x(t) = bt2 − ct3, where b = 2.40 m/s2 and c = 0.120 m/s3. (b) Calculate the instantaneous velocity of the car at t = 0, t = 5.0 s, and t = 10.0 s.

Video transcript

Welcome back everybody. We are given ambulance that is traveling towards the hospital and its position is given by this function at any given time where B is equal to 3.10 and the constant C is equal to 0.147. We are asked to find the instantaneous velocity when t equals zero equals 2.5 E equals 7.5. So first and foremost I'm actually gonna go ahead and write out our position but plug in those constants. So we have position with respect to T is 3.1 E squared minus 0.147. So now how are we going to get instantaneous velocity? Well, the derivative of position with respect to time is velocity with respect to time. So we have to take the derivative of this function right here and we'll do it component by component. So the derivative Position back the time is going to be delivered with this guy. And since this exponents here we're gonna use the power rule. So this is gonna be two times three one T minus this guy. Same thing. Got the exponent here. It's three times 0.147 B squared. So this is equal to our velocity with respect to time. So now we can go ahead and find the velocity at the three different times. So let's start with the velocity and T. Is equal to zero and we just plug in zero wherever T. Is. This is gonna be two times 3. times zero minus three times 0. zero squared this of course just equals zero. Now go ahead and find the instantaneous philosophy at 2.5 seconds Would equal two times 3. Times 2.5 -3 times 0.147 times 2.5 squared. Which when you plug into your calculator, you get to 12.7 m per second. Now we just got one more to go. Let's go ahead and find velocity At 7.5 seconds. Going to equal two times 3. I'm 7.5 minus three times 0.147 times 7.5 squared. Plugging this into our calculator, we get 21.7 m per second, which means that we have found instantaneous philosophy at all three times, which corresponds to our answer choice of Thank you guys so much for watching. Hope this video helped and we'll see you all in the next one.