Test: Kinematics and Kinetics - Mechanical Engineering MCQ

The velocity of the particle is defined as the rate of change of its displacement and can be expressed as
v = dx/dt
whereas, the speed of the particle at any instance is given as the rate of change of its distance.
for maximum velocity 
dv/dt = 0
Calculation:
Given :
Position: x= 6t² - t³
dx/dt = V = 12t - 3t²   ...(i)
and for maximum velocity 
dv/dt = 0,
dv/dt = 12-6t = 0 or 6t = 12 or t = 2 sec
Substitute this value of t in the (i) 
we get v = 12× 2 - 3× 2² = 12 m/sec

Linear acceleration is defined as the rate of change of linear velocity of a body with respect to the time.
i.e a = v/t
and unit of velocity is m/s
so, unit of linear acceleration becomes m/s².

Angular velocity is defined as the rate of change of angular displacement with respect to time. It is usually expressed by a Greek letter ω (omega).
Mathematically, angular velocity,
ω = dθ/dt
If a body is rotating at the rate of N r.p.m. (revolutions per minute), then its angular velocity,
ω = 2πΝ / 60 rad/s

If a ball is dropped from a height ho on a horizontal floor, then it strikes with the floor with a speed,

and it rebounds from the floor with a speed, 

Where, Coefficient of restitution = 
First height of rebound, 


Therefore,

The velocity of ball after n^th rebound; V_n = eⁿV_o
The height of ball  after n^th rebound,  h_n = e²ⁿh_o
Calculation: 
Given: h_o = 3 m., e = 0.5, h₁ = ?
We know that, height after first rebound: 
h₁ = (0.5)² × 3 = 0.75 m.

When two bodies impact, the Momentum is conserved, i.e.
m₁u₁ + m₂u₂ = m₁v₁ +  m₂v₂
Properties of different types of collision are given in the table below:

Perfectly elastic impact

• Momentum is conserved, m₁u₁ + m₂u₂ = m₁v₁ +  m₂v₂
• Kinetic Energy is conserved, 
• The two bodies separate after the impact
• There is no permanent deformation in the bodies during the impact.
• In a perfectly elastic collision between equal masses of two bodies, velocities exchange on impact.
• The relative velocity of the two bodies after impact is equal and opposite to the relative velocity of the two bodies before impact.

Plastic impact

• The two bodies move together with a common velocity after the impact
• Moment is conserved, i.e. m₁u₁ + m₂u₂ = (m₁ + m₂) v₀
• where v₀ is the common velocity after impact.
• There is a loss in Kinetic energy after the impact.
• The coefficient of restitution is zero.

If the displacement is along a circular path, then the direction of linear velocity at any instant is along the tangent at that point.
therefore, the linear velocity will be ω.r

The acceleration of a particle at any instant moving along a circular path in a direction tangential to that instant, is known as tangential component of acceleration or tangential acceleration.

Coefficient of restitution (e):
Coefficient of restitution or coefficient of the resilience of a collision is defined as the ratio of the relative velocity of separation after the collision to the relative velocity of the approach before the collision.


A perfectly inelastic collision also called a perfectly plastic collision is a limiting case of inelastic collision in which the two bodies stick together after impact.
Since both bodies stick together in a perfectly plastic collision. Therefore v_A = v_B, Thus e = 0
Properties of different types of collision are given in the table below:

When the Force F is suddenly removed, then due to W, the rod is in rotating condition with angular acceleration α
Thus the equation of motion:

The acceleration of a particle at any instant moving along a circular path in a direction normal to the tangent at that instant and directed towards the centre of the circular path is known as normal component of the acceleration or normal acceleration. It is also called radial or centripetal acceleration.