 Let's solve a couple of questions on change in momentum and velocity from force time graphs. Here we have a child who pushes a 2 kilogram swing that is initially at rest. The net force on the swing over time is shown below. Here we have the graph force versus time graph, force and Newton's time in milliseconds. Question is to figure out the swings speed at 30 milliseconds and we can round our answers to two significant figures and time is in milliseconds that you can already see. Alright before I get into this, why don't you pause the video and first attempt this one on your own. Alright, hopefully you have given this a shot. Now we can think about this by thinking about what is momentum really? How did we define momentum? We know one way of defining it was b equals to mv and we can write this as change in momentum. This is equal to m delta v. But there was also one other way of defining, in fact the other way of defining Newton's second law. Still now we were defining Newton's second law as f equals to ma. But now we know that force is really just the rate of change of momentum. This is really just delta p divided by delta t and if we take delta t on the left hand side we can write delta p as f into delta t which means that for any force versus time graph if we think about the area and the curve that will give us a change in momentum. And over here we need to think about the swing speed at t equals to 30 milliseconds. So if we know the final momentum that is momentum at 30 seconds, if we know mvf the final momentum, we know the mass and we can then figure out vf if we know the final momentum. We know initially the momentum is zero because it is starting from rest and there is no velocity. So when we think about delta p, delta p is really just final minus initial. So if we know the final momentum, we can then figure out the final speed. Now let's come to the graph. We can see that the force is increasing with time which means that there will be a constant change in speed. There will be a constant change in speed of the swing as the force is increasing in time constantly. And we need to only see up till 30 milliseconds. So let's do that up till 30 milliseconds. Let's try and see what the area is. So this is 30 and on the y-axis, approximately we can estimate that it is touching the y-axis in between 20 and 40 at the 30th mark and when force is 30. So if we try to find the area of this triangle, the area of this triangle that would be delta p. This is the area of the triangle is giving us a change in momentum that is half into base into height. So this is half into 30 into 30 and this comes out to be equal to 450 Newton into millisecond. But this is a change in momentum and we can expand change in momentum. We can expand change in momentum using this relation right here. We just saw that change in momentum delta p is also equal to m delta v. So we can write this, we can write this as m delta v or we can write this as mass into final velocity minus initial velocity when you multiply m with vf and vi that will be final momentum minus initial momentum and initially we know that this is at rest. So vi is just 0 m into velocity m is 2. So 2vf this is equal to 450 this is equal to 450 Newton into milliseconds. We can change milliseconds to seconds because that is we need to report the answer in meters per second. So 450 Newton into milliseconds we know that 1 millisecond this is equal to 10 to the power minus 3 seconds. So 450 Newton into milliseconds that would be 450 divided by 1000 and this is 0.45 Newton's second. So instead of 450 we can write 0.45 Newton's seconds and when we do that when we do that vf comes out to be equal to 0.45 divided by 2 and that is 0.225 meters per second. We need to round the answer to two significant figures. So we can write 0.225 as 0.23 this is 0.23 meters per second. So the swing speed at 30 millisecond is 0.23 meters per second. Now let's move on to our next question. Here we have a squirrel who pushes a 1.2 kilogram apple that is initially at rest on the ground and the net force on the apple varies with time as shown below. What is the apple's change in momentum between 0 millisecond and 20 milliseconds? We need to again round the answers to two significant figures and time is in milliseconds but we need to report the final answer in meters per second. Alright again pause the video and give this one a try. Hopefully you have given this one a try as well. So again following the same logic as the last question we note that delta P this is equal to F into delta T that is the area under the force time graph will give us a change in momentum and over here we need to figure out the change in momentum between 0 and 20 milliseconds that's basically the area of this entire triangle. So if you try to figure out the area this is half into base into height here base is 20 so this is half into 20 into height which is which is just minus 20 which is minus 20. So this comes out to be equal to minus 200 Newton into milliseconds and we can change milliseconds to seconds by dividing this by 1000 so this comes out to be delta P this comes out to be equal to minus 200 divided by 1000. So this really is 0.2 minus 0.2 Newton second and we can write this unit we can write this unit in this manner we know that Newton is Newton is really kilogram into meters per second square and this has been multiplied with a second so this gets crossed off and ultimately this becomes minus 0.2 kilogram meters per second. Just a quick recall why is Newton kilogram into meters per second square because Newton force force is really ma mass is kilogram and acceleration is meters per second square. So the change in momentum change in momentum is minus 0.2 minus 0.2 kilogram meters per second.