**Question:**

**Give examples of a one-dimensional motion where**

**(a) the particle moving along positive x-direction comes to rest periodically and moves forward**

**(b) the particle moving along positive x-direction comes to rest periodically and moves backwardπ**

**Solution:**

When an equation has sine and cosine functions, the nature is periodic.

(a) When the particle is moving in positive x-direction, it is given as t > sin t

When the displacement is as a function of time, it is given as x(t) = t – sin t

When the equation is differentiated with respect to time, we get

Velocity v(t) = dx(t)/dt = 1 – cos t

Differentiating the above equation again with respect to time, we get

Acceleration, a(t) = dv/ dt = sin y

When t = 0, x(t) = 0

When t = π, x(t) = π > 0

When t = 0, x(t) = 2 π > 0

(b) The equation is given as

x(t) = sin t

v = (d/dt)x(t) = cos t

a = dv/dt = -sin t

At t = 0, x = 0, v = 1 and a = 0

At t = π/2, x = 1, v = 0 and a = -1

At t = π, x = 0, v = -1, and a = 0

At t = 3 π/2, x = -1, v = 0 and a = 1

Therefore, it can be said that when the particle is moving along the positive x-direction, the particle comes to rest periodically and moves backward. When the displacement and velocity is involved, that is sin t and cos t, the equations are represented periodic in nature.