Question

A particle moves in a straight line with retardation proportional to its displacement .its loss of kinetic energy for any displacement X is proportional to what

Answer 1

"Acceleration a = dv/dt,

Velocity v = dx/dt, where ‘x’ is position and ‘dx’ is displacement.

Take the ratio of the two equations. We get:

a/v = (dv/dt) ÷ (dx/dt)

Cancel the dt, rearrange the terms….

vdv = adx —————(1)

The question says that -a ∝ x.

-a = kx, where k is a constant of proportionality. Let’s substitute this in equation (1).

-vdv = (k)xdx

Multiply both sides with mass m.

-mvdv = (km)xdx

-mvdv = -d(½mv²) = -dK and xdx = d(½x²)

-dK = km * d (½x²)

Integrate both sides.

-∫mvdv = ∫ (km)xdx

-ΔK = (km) (½x²)

This means that -ΔK ∝ x²

Therefore, -ΔK is the loss in Kinetic energy.

"

Answer 2

see the attachment for the answer