A particle starts from rest at time t = 0 and moves along the x axis.If the net force on it is proportional to t, its kinetic energy is proportional to:

Question

Quizplus · Accepted Answer

When the net force on a particle is proportional to t, we can write the force as F = kt, where k is a constant. By Newton's Second Law, we know that F = ma, or kt = ma. Rearranging, we get a = (k/m)t.

We can integrate this acceleration function to find the particle's velocity as a function of time:

v(t) = a∫dt = (k/m)∫t dt = (k/m)(t^2/2) + C

Using the initial condition that the particle is at rest at t = 0, we can solve for C:

v(0) = 0 = (k/m)(0^2/2) + C 
C = 0

So the particle's velocity as a function of time is:

v(t) = (k/m)(t^2/2)

Now we can find the kinetic energy of the particle as a function of time:

KE(t) = (1/2)mv(t)^2 
KE(t) = (1/2)mk^2(t^4/4m^2) 
KE(t) = (k^2/8m)(t^4)

Since k and m are constants, we see that the kinetic energy is proportional to t^4. Therefore, the answer is (C) t^(1/4).

A Particle Starts from Rest at Time T = 0