The momentum p of a particle changes with time t according to the relation $\frac{dp}{dt}$ = (10 N) + $\left(2\frac{N}{s}\right)$t. If the momentum is zero at t= 0, what will the momentum be at t = 10 s?

Step 1, Given data

$\frac{dp}{dt}=\left(10N\right)+\left(2\frac{N}{S}\right)$ t

Momentum is zero at t= 0

Momentum at t =10 sec will be,

Step 2, Finding the momentum

From the given equation

$dp=(10N+2N{s}^{-1}t)dt$

Integrating both sides to get the momentum

${\int }_0^{p}dp={\int }_0^{10}10dt+{\int }_0^{10}\left(2tdt\right)$

P = ${[10t+2\frac{t^2}{2}]}_0^{10}$

P = 10 $\times$ 10-0 +100

P = 100+100= 200 kg.$\frac{m}{s}$

Hence the momentum at 10 sec is 200 kg.m/s