in this problem I'll talk about the impulse momentum theorem and here the force is not a constant it does change at the time okay so this force is a time-dependent force and this force is now being applied onto a mass and if the man mass starts from and the rest what is the speed of the object after time for T equals to four second so let me explain this so let's say you have a mass here and this force is now being applied and this is at rest so the initial velocity is at rest and you can see the force is time-dependent it does a value at that time so after four second the object is at this location you need to find out what is the speed of this object and how we're going to do it so the simple way of doing this one is using the conservation and using the impulse momentum theorem that's the straightforward way of doing this one what is the impulse momentum theorem tells you the impulse tells you the impulse is equal to the change in momentum and the formula for calculating impulse is this F time F is the force times time and then taking the determinant the integration over the entire interval that gives you decomposed or in other words how can you understand impulse how long does a force act on to the system the product the force and the time is the impulse for the amount of the time of force X onto a system is the impulse and this is exactly equal to the change in the momentum so now the force is given so I'm just writing down the force DT is the time and change in momentum is the final momentum this is a final momentum and that's the initial momentum so this is the final momentum and this is the initial momentum FS stands for the final and the highest ann's for the condition so the mass is still kilogram the velocity we do not know it the mass is again to Club Ram but initial velocity is 0 so we have 0 here and then not we need to integrate this one and I just like to remind you that the integration let's say you have a function X to the N DX and if you integrate what you get is X to the n plus 1 divided by n plus 1 so you just need to keep this in mind okay the other thing you have to notice the the exponent has increased by 1 1 and then you have n plus 1 here and the torque so if we now integrate the 20 is a constant so it will be now TQ divided by 3 again 50 T squared 1 up and divided by 2 and the time the time is from here the time is T plus to 0 second and here time is equal to 4 second so the limit is from time equals to 0 to time goes to 4 second and this is simply equal to 2 times V F the first wheeled the plug time equals to 4 second so 20 divided by 3 42 plus 50 divided by 2 is 25 T is now 40 square x is equal to 2 V F and if you solve it what you get is 413 meter per second so we now by using the impulse momentum theorem found out what is the speed of the the particle when a force on reading for the time dependent force X onto it so this is it from this impulse momentum theorem and again do not forget to like share and subscribe