Question

The kinetic energy of an object with a mass of `2 kg` constantly changes from `32 J` to `136 J` over `4 s`. What is the impulse on the object at `1 s`?

Answer 1

I am facing this type of question but no one has answered it yet.
Probably the question is not clear enough as impulse at a particular time is wanted here , which is not possible by definition

Now I place here a probable solution considering that the impulse at a particular time (say Tsec) means

Either (1) It is the change in momentum during `t=0 to t = T sec` (this is by definition )

Or (2) It is the momentum of the body at `T s` ( assumed )

In our question the KE of an object of mass `m = 2kg` constantly changes from `32J to 135J` in `4s`. So during this time KE vs time graph will be straight line having a + slope
So slope `=(Delta E)/(Deltat)= (E_f-E_i)/(t-0)=(136-32)/(4-0)=26`

If the velocity of the object is v at `t` th sec Then
we can write
`(E_t-E_i)/(t-0)=26`
`=>(1/2xx2v^2-32)/(t-0)=26`
`=>v^2=26t+32`
`=>v=sqrt(26t+32)`
So the momentum at `t=0s`
`p_0=2xxv_0 =2xxsqrt32kgm/s=`
So the momentum at `t=1s`
`p_1=2xxv_1 =2xxsqrt(26xx1+32)=2sqrt58kgm/s`

Hence impulse during 1s = `p_1-p_0=2(sqrt58-sqrt32)kgm/s`

OR otherwise impulse at 1s is momentum at is `=p_1 =2sqrt58kgm/s`