Impulse and Momentum - Free MCQ Practice Test with solutions, GATE Civil

• Momentum is conserved in all collisions.
• In elastic collision, kinetic energy is also conserved.
• In inelastic collision, kinetic energy is not conserved. In perfectly inelastic collision, objects stick together after the collision.

Perfectly elastic collision:
If law of conservation of momentum and that of kinetic energy hold good during the collision.

Inelastic collision:
The law of conservation of momentum holds good during a collision while that of kinetic energy is not.
Coefficient of restitution (e)

• For perfectly elastic collision, e = 1
• For inelastic collision, e < 1
• For perfectly inelastic collision, e = 0

This is the perfectly inelastic collision/impact of the case of two bodies/balls as both the balls stick after the collision.
Perfect Inelastic Collision:

• Two bodies move together with the same velocity.
• The coefficient of Restitution will be 0.
• Momentum is Conserved.
• Kinetic Energy will not be Conserved.

​Calculation:
Given:
Let two balls A and B have mass m_A, m_B respectively, and their initial velocities are u_A and u_B. After the collision, they will move with the same velocity, v_o.
Given that mass of both balls are same.
So  m_A = m_B = m
u_A = V,  u_B = 0
From the Concept of Momentum Conservation:
m_Au_A + m_Bu_B = (m+m)v_o
mV = 2mv_o
v_o = V/2
Both the balls after impact will move with velocity v/2.

The correct answer is Both total kinetic energy and total linear momentum

• A perfectly inelastic collision is a hypothetical collision in which two objects collide and energy is not wasted and momentum is conserved.
• While Perfectly elastic collision happens then the object’s shape remains unchanged or not deformed.
• Most of the collisions in daily life are inelastic in nature.

Important Points

Linear momentum:

• It is the product of the mass and velocity of an object.
• Like velocity, linear momentum is a vector quantity possessing a direction as well as magnitude.

P = mv
where P = momentum and v = velovity.
Calculation:
Given:
Mass = 9 kg, P = 63 kg-m/sec
P = mv
63 = 9 × v
v = 7 m/s

• Kinetic energy (K.E): The energy possessed by a body by the virtue of its motion is called kinetic energy.

The expression for kinetic energy is given by:
KE = (1/2)mv²
Where m = mass of the body and v = velocity of the body

• Momentum (p): The product of mass and velocity is called momentum.

Momentum (p) = mass (m) × velocity (v)
The relationship between the kinetic energy and Linear momentum is given by:
As we know,
KE = (1/2)mv²
Divide numerator and denominator by m, we get

Calculation:
Let initial Kinetic energy = KE₁ = E
Given that:
Final kinetic energy (K.E₂) = K.E₁ + 300 % of KE₁ = E + 3E = 4E
The relation between the momentum and the kinetic energy is given by:

Final momentum (P') will be:

Increase in momentum (ΔP) = P' - P = 2P - P = P
% Increase = 

To topple the block we need an impulse force such that the weight will generate a clockwise moment about point O.
This will happen only when a minimum impulse that can rotate the block such that the Centre of gravity of the body will pass through the point O or beyond.

h = 0.4m, b = 0.3 m, m =1kg

h' = OG' - h/2 = 0.25 - 0.2 = 0.05 m
Due to force F, the block will topple about A
So By energy balance

ω² = 2 × 12 × 9.81 × 0.05
ω = 3.43 rad/s
Now,
Angular impulse = change in angular momentum
I_F × d = I_o × (ω_i - ω_f)
I_F × 0.3 = 1/12 × ( 3.43 - 0)
I_F = 0.953 Ns

Newton’s Second law of motion: The rate of change of momentum of any object is directly proportional to the applied force on the body.

F × ΔT  = ΔP
Where Δ P = Change in momentum and Δ T = change in time taken

• The above equation is known as Impulse Momentum equation and states that the impulse or force intensity is equal to change in momentum.
• According to the impulse-momentum equation, the change in momentum of an object depends on both the net force acting on the object and duration of the net force.

Impulse (J): 

• It is defined as the integral of force with respect to time.
• It is also defined as a change in the linear moment (P) with respect to time.
• It is a vector quantity.

J = F × dt = ΔP 
Calculation:
Given:
m = 1.0 kg, v₁ = 25 m/s, v₂ = 10 m/s
Impulse = change in momentum
J = ΔP
J = m(v₁ - v₂)
J = 1.0 × (25 - (-10))
= 1× (25) - 1 ×(-10) = 35 N - s 
∴ J = 35 N-s

The correct answer is 3 units.

• Newton's 2nd law of motion says the rate change of momentum is called force.
• The equation for the change in momentum is F ×  t = m ×  Δv
• If the momentum of a body increases from 10 units to 25 units in 5 seconds.
• Then the change in the momentum is 25 – 10 = 15 units in time 5 seconds.
• So the Force extract from this is 15/5 = 3 Newton.

Key Points

• Momentum is a vector quantity that is the product of the mass (weight) of a particle and its velocity (speed).
  • SI unit: kilogram meter per second (kg⋅m/s).
• A spinning object has angular momentum.
• An object travelling with a velocity has linear momentum.
• A change in momentum may result from an acceleration or a force or an impulse.

• Momentum is conserved in all collisions.
• In elastic collision, kinetic energy is also conserved.
• In an inelastic collision, kinetic energy is not conserved. In a perfectly inelastic collision, objects stick together after the collision.

Conservation of momentum:
m₁v₁ + m₂v₂ = (m₁ + m₂) v_f
Loss of kinetic energy:

Calculation:
Given:
m₁ = m₂ = 10 kg, v₁ = 15 m/s, v₂ = 5 m/s
Loss of kinetic energy: