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# Worked example: Motion problems with derivatives

AP.CALC:
CHA‑3 (EU)
,
CHA‑3.B (LO)
,
CHA‑3.B.1 (EK)

## Video transcript

a particle moves along the x-axis the function X of T gives the particles position at at any time T is greater than or equal to zero and they give us X of T right over here what is the particles 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 is 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 4t squared with respect to T is negative eight T and derivative of 3t with respect to T is +3 derivative of a constant doesn't change with respect to time so that's just a 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 to wherever we see the T's so it's gonna be three times four three times two squared so it's 12 minus 8 times two minus 16 plus three which is equal to negative one and you might say negative 1 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 1 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 is negative that means that X is decreasing or we are moving to the left what is the particles 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 is a function of time is 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 3t squared with respect to T is 60 derivative of negative 80 with respect to T is a minus 8 and derivative constants zero so it's just going to be 60 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 asked us what is the direction of the particles motion at T equals 2 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 particles speed increasing and 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 particles 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 2 our velocity is negative 1 if the units were meters and second it would be negative 1 meters per second but our speed would just be 1 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 I'll have different signs well that means that your speed is decreasing the magnitude of your velocity would become mean less so let's look at our velocity at time T equals 3 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 two-six saw 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