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
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.