There's a miniature rocket ship and it's full of tiny aliens that just got done investigating a new moon with lunar pools and all kinds of organic new life-forms but they're done investigating so they're going to blast off and take their findings home to tell all their friends let's say at some moment during their ascent they're .
Moving at 4 meters per second and their tiny aliens their spaceship is only 2.9 kilograms but they need to know are they going to be able to get off this moon or not so they've got to pay attention to their speed but instead of using a speedometer the clever aliens they use a force versus time graph so on their dashboard that we've got a force versus .
Time graph and it tells them what the net force is on them so let's say this is the net force not just any force this is the total force on them from rocket boosters and the force of gravity and whatever other forces there might be they've got too advanced for sensors I mean come on they can determine their net force let's say and it gives them .
This force as a function of time but they want to know what is their velocity going to be after 9 seconds so they check their force versus time readout and this is the graph they get and now they can determine it and here's how they do it they say alright there's a force a net force of 3 Newtons acting for the first 4 seconds so that during .
This entire first 4 seconds there's a constant force of 3 Newtons and every alien worth his weight knows that force the net force multiplied by the time duration during which that force is applied gives you the net impulse so this gives us the net impulse if we take this constant 3 Newtons that acts multiplied by four seconds during which .
It acts we get that there's an impulse of 12 Newton seconds now you might be like wait who cares about Newton's seconds here I want the velocity I don't care about the force and the time I want to know the velocity at 9 seconds but they teach you at the alien space Academy that the net impulse is not only equal to the net force times the time .
It's also equal to the change in momentum of the object that the force was exerted on and this is good we know the mass of the object we want to know something about velocity so we know momentum is M times V this net impulse is going to help us get there but this 12 Newton seconds was only for the first 4 seconds how do we .
Figure it out for the next three seconds leave this during the next 3 seconds there's not a constant force this force is varying the force is getting smaller so how do I do this the force isn't a constant value so I can't just simply take Force Times time because I mean what force do I pick so we're going to use a trick we're going to use a trick .
Because if you notice for this first section for the first four seconds we took the force and multiplied by the time interval four seconds so what we did really is we just took the height of this rectangle times the width of this rectangle and that gives us the area of this rectangle so what we really did is we found the area under the force versus .
Time graph that gave us the impulse and that's not a coincidence the impulse equals the area under a force versus time graph and this is extremely useful to know because now in this section where the force was varying we can still use this we can just find the impulse by determining the area under that curve and by area under the .
Curve we mean from the line curve in general to the x-axis which in this case the x-axis is of the time axis so let's do this we found the impulse for this first section that was 12 Newton seconds now we can find the impulse for this next section by just determining the area so this is a triangle we'll do one half base the base is 1 2 3 seconds and .
The height is still 3 Newton's so we get a net impulse of 4.5 Newton seconds so if you've got one more section to go but this one's a little weird this one's located the area is located below the time axis so this is still a triangle but since the forces are negative this is going to count as a negative net impulse so when the area lies above the .
Time axis it counts as a positive impulse and when the area lies below the time axis it counts as a negative net impulse so how much negative net impulse we still find the area so the area of a triangle again is going to be 1/2 the base this time is 2 seconds and the height is negative 2 so negative 2 Newtons which gives us a net in Pole .
Of negative to newton-seconds and now we can figure out the velocity of this spaceship at nine seconds so assuming that this force readout started at this moment right over here at T equals zero seconds was the moment when it was going four meters per second then we can just say the total net impulse should equal the total change in momentum of the .
Spaceship and we can find the total net impulse by just adding up all the individual impulses so during the first four seconds there was 12 Newton seconds of impulse during the next three seconds there was 4.5 Newton seconds of impulse and during this last portion there was negative 2 Newton seconds of impulse which if you add all those up 12 plus .
4.5 plus negative 2 you're going to get positive 14.5 Newton seconds of impulse that's good news for our alien buddies over here they need to get off this moon which means they need positive impulse upward impulse they got some positive impulse let's see what their final velocity was we know that Delta P is the change in momentum so this is final .
Momentum minus initial momentum which we could write as mass times V final minus mass times V initial now if this were an earth rocket this would be hard because earth rockets using earth technology eject fuel at a huge rate out the back end and that loses mass that means this mass isn't going to stay constant so earth Rockets essentially push fuel down .
Which causes an equal and opposite force back on the rocket upward but if you're losing mass this mass doesn't stay constant and this whole process is a lot harder because M final and M initial aren't going to be the same maybe this is where the phrase it's not rocket science comes from because rocket science is a little harder when that .
Mass changes so let's just say these clever aliens can eject only a little bit of fuel they do so you might say how well there's got to be a certain amount of momentum right that they eject to give themselves momentum up but let's say they can eject only a small mass at a huge speed so there's not much fuel that they're losing the fuel that they .
Eject is ejected at the enormous speed so that they get their momentum upward but they lose almost no mass and that lets us solve this problem a soon the mass is constant so if we do that if we assume the mass is constant we get positive 14.5 Newton's seconds equals we can pull the mass out the masses of constant so I could just write it as M .
Times V final minus V initial since I can pull out a common factor of M which means I can write this as 2.9 kilograms multiplied by the final velocity after 9 seconds and I know it's after 9 seconds because I added up all the impulse during the 9 seconds minus the initial velocity was 4 meters per second so if I divide both sides by 2.9 kilograms 14.5 .
Over 2.9 is 5 and that will be Newton seconds over kilograms which has units of meters per second and it's positive that's going to equal the left over over here is going to be V final minus 4 meters per second and now finally if I add 4 meters per second to both sides I get that V final somewhere up here the final of this rocket is going to be 9 .
Meters per second so after 9 seconds it ended up going 9 meters per second that's just a numerical coincidence the way you find it so recapping the way we did this we found the area under the curve because the area under a curve under our force versus time graph represents the impulse on the object we found it for the entire trip noting that .
Underneath the time axis when this curve goes underneath the time axis the net impulse is going to be negative we added up all the net impulse we set it equal to the change in momentum we plugged in our values and we solved for our final velocity after 9 seconds