In this problem, we are given that a particle is moving in a straight line with an initial velocity u, and its retardation at any instant is proportional to the displacement. We need to find the displacement of the particle before it comes to rest.



Let the displacement of the particle at any instant be x, and let its retardation at that instant be a. We are given that a is proportional to x, which means:

a = kx

where k is a constant of proportionality.

We know that acceleration is the rate of change of velocity with time, and retardation is negative acceleration. Therefore, we can write:

a = d²x/dt²

where d²x/dt² is the second derivative of displacement with respect to time. Substituting the value of a, we get:

d²x/dt² = kx

This is a second-order linear differential equation with constant coefficients. The solution of this equation is:

x = A cos(ωt) + B sin(ωt)

where A and B are constants of integration, and ω = sqrt(k).

Now, we need to apply the initial conditions to find the values of A and B. The initial velocity of the particle is u, which means:

dx/dt = u

Substituting x = A cos(ωt) + B sin(ωt), we get:

-dAω sin(ωt) + dBω cos(ωt) = u

At t = 0, the displacement of the particle is zero, which means:

x = 0

Substituting x = A cos(ωt) + B sin(ωt), we get:

A = 0

Now, we can solve for B and get:

B = u/ω

Substituting A = 0 and B = u/ω in x = A cos(ωt) + B sin(ωt), we get:

x = (u/ω) sin(ωt)

The particle will come to rest when its velocity becomes zero. Therefore, we need to find the time when the velocity of the particle becomes zero. We know that:

dx/dt = u cos(ωt)

The velocity becomes zero when cos(ωt) = 0, which means:

ωt = pi/2

t = pi/2ω

Substituting t = pi/2ω in x = (u/ω) sin(ωt), we get:

x = u/ω

Substituting ω = sqrt(k), we get:

x = u/sqrt(k)

We are given that the retardation at unit displacement is pi, which means:

a = pi

Substituting a = kx, we get:

pi = k(u/sqrt(k))

k = pi²/u²

Substituting k in x = u/sqrt(k), we get:

x = u/pi

Therefore, the displacement of the particle before coming to rest is u/pi.



In this problem, we found the displacement of a particle moving in a straight line with an initial velocity u and retardation proportional to