Question

A particle of mass m moves with a constant acceleration a. If the initial position vector and velocity of the particle are r0 and v0 respectively, use energy argument to show that its speed v at any time satisfies the equation v^2 =v 0^2+2a⋅(r−r0 ).

Answer 1

Using the work-energy principle, the work done by the force accelerating a particle changes its kinetic energy. Equating work done to the change in kinetic energy and simplifying provides the equation for the particle's speed as a function of time.

Step-by-step explanation:

To show that the speed v of a particle with mass m moving with constant acceleration a satisfies the equation v^2 = v0^2 + 2a · (r - r0), we can use the work-energy principle. The work done by the force responsible for the acceleration results in a change in kinetic energy. The work done W by the force is equal to the dot product of the force and the displacement, W = F · (r - r0). Since F = ma, this becomes W = ma · (r - r0). The change in kinetic energy ΔK is K_final - K_initial, which is rac{1}{2}m(v^2) - rac{1}{2}m(v0^2). By the work-energy principle, W = ΔK, so ma · (r - r0) = rac{1}{2}m(v^2) - rac{1}{2}m(v0^2). Simplifying this equation gives us the desired result of the particle's speed at any time fulfilling the equation v^2 = v0^2 + 2a · (r - r0).