Question

A particle of mass m moves in a straight line with retardation proportional to it's displacement. Find the expression for loss of kinetic energy for any displacement x.

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

mass = m
Path = Straight Line (Linear Curve)
a [Directly Proportional to] -x
a = -k•x (k = constant of proportionality)
Integrating on both sides with proper limits,
v = -k • x²/2
In S.I., k = 1, Hence, v = -x²/2, v² = x⁴/4
K = mv²/2
K = mx⁴/8.