The net force F = 4t.ex + 2.ey is acting on a body with mass m = 1 (kg) that is moving

Question

The net force F = 4tex + 2ey is acting on a body with mass m = 1 (kg) that is moving. v(0) = 0 is the initial velocity. Find the kinetic energy of the body K(t) as a function of time and the body's momentum p(t) as a function of time.

Answer 1

I will speculate that your notation means

F(t) = 4t.e_x + 2.e_y

where e_x is the unit vector in the x-direction and e_y is the unit vector in the y-direction.

Let me first calculate the momentum p(t).

By Newton's Second Law

F = dp/dt

So

p = ∫Fdt

= ∫(4t.e_x + 2.e_y)dt = 2t².e_x + 2t.e_y + C

The integration constant C is determined from the additional information that v(0) = 0, i.e.,

p(0) = 0

Therefore C = 0

The kinetic energy

K(t) = m|v|²/2

Using

v = p/m

we rewrite

K(t) = m|p/m|²/2

= |p|²/2m

= |2t².e_x + 2t.e_y|²/2m

= (4t⁴ + 4t²)/2m

= 2t⁴ + 2t²