CALC A 500 g particle has velocity vx = −5.0 m/s at t=−2 s .Force Fx= (4−t²) N , where t is in s, is exerted on the particle between t=−2 s and t=2 s . This force increases from 0 N at t=−2 s to 4 N at t=0 s and then back to 0 N at t=2 s . What is the particle's velocity at t=2 s?

Video transcript

Hey, everyone. So this problem is dealing with variable forces. Let's see what it's asking us consider a 300 g object that has an initial velocity of two m per second in the positive X direction at time equals zero seconds. If a force F sub X is equal to six plus two T newtons, where T is in seconds is applied to the object in the X direction between T equals zero and T equals four seconds. How fast will the object be in the X direction at T equals four seconds? Our multiple choice answers here are a 415 m per second. B 531 m per second. C 135 m per second or D m per second. So the first step in this problem is recalling that the impulse momentum theorem is given by force multiplied by time is equal to mass multiplied by the change in velocity. Now, we can integrate our force equation with respect to time in order to solve for the um seed at this final time of four seconds. So that looks like the integral of force multiplied by D T is equal to mass. And then delta B is the same as V F minus V I. And so when we integrate from 0 to 4 seconds of our force function six plus two T, the units are newtons D T and that's equal to, again, mass multiplied by delta feed. So we take that integral and we have T squared plus six T from 0 to seconds units of oops, sorry. Um It's all right. So that's just yeah, squared plus six T from 0 to 4 seconds is equal to mass times that delta V V F minus V I. And now we can divide both sides by mass as we are um solving for this T squared plus six T. So when we do the integral, it'll be four seconds plugged in minus zero seconds, but zero, both of those terms go to zero. So simply be four seconds squared plus six multiplied by four seconds. And then when we divide out that mass from the other side, mass is 0.3 kg. It's given to a problem. And then finally, we can add our initial speed again from the problem two m per second. And that is how we get V F. So once we isolate that variable plug that all into our calculator, we get 135 m per second. So that's the final answer for this one. And that aligns with answer choice. That's all we have for this video. We'll see you in the next one.