Question

A particle of mass m is accelerated along the positive x-direction by a constant force; It starts from rest at the origin of an inertial frame. A second reference frame moves with constant speed V0 along the negative x-direction ; initially, the frame coincide

a) FInd the velocity and position of the particles as a function of time in both reference frames
b) Find the work done by the force during a time interval t in both frames

Answer 1

Answer on Question #57842-Physics-Mechanics

A particle of mass $m$ is accelerated along the positive $x$-direction by a constant force; it starts from rest at the origin of an inertial frame. A second reference frame moves with constant speed $V0$ along the negative $x$-direction; initially, the frame coincides with an inertial frame.

a) Find the velocity and position of the particle as a function of time in both reference frames

b) Find the work done by the force during a time interval $t$ in both frames

Solution

a) The velocity of the particle in first frame:

$$v = 0 + a t = a t.$$

The position of the particle in first frame:

$$s = \int_{0}^{t} v \, dt = \frac{a t^{2}}{2}.$$

The velocity of the particle in second frame:

$$v' = v_0 + a t.$$

The position of the particle in second frame:

$$s' = \int_{0}^{t} v' \, dt = v_0 t + \frac{a t^{2}}{2}.$$

b) The work done by the force during a time interval $t$ in the first frame is

$$W = m a s = m a \frac{a t^{2}}{2} = \frac{m a^{2} t^{2}}{2}.$$

The work done by the force during a time interval $t$ in the second frame is

$$W' = m a s' = m a \left(v_0 t + \frac{a t^{2}}{2}\right) = m a v_0 t + \frac{m a^{2} t^{2}}{2}.$$

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