Question

Block A, having a mass m, is released from rest falls a distance h and strikes the plate B having a mass 2m. If the coefficient of restitution between A and B is e, determine the velocity of the plate just after collision. The spring has a stiffness k.

Answer 1

Concepts and reason

Law of conservation of energy:

According to which, energy can neither be created nor be destroyed, but it can be transformed from one form to another form.

Linear momentum:

Linear momentum of a particle refers to the product of mass and the velocity of the particle. It is denoted by and the unit is .

Law of conservation of momentum:

In an isolated system, when two objects undergo collision with each other, then the total momentum before the collision is equal to the total momentum after the collision such that the momentum between two objects is conserved. The sum of the initial linear momentum of a particle for a process is equal to the sum of the final linear momentum of the particle.

Here, the initial linear momentum of mass is , the initial linear momentum of mass is $(m_v2); $, the final linear momentum of mass is , and the final linear momentum of mass is .

Potential energy:

It refers to the energy contained by the body held at a vertical position with a reference to the surface.

Kinetic energy:

It refers to the energy possessed by the body that is said to be in a motion with a given velocity.

Coefficient of Restitution:

It refers to the ratio of the relative velocity of two objects before impact to the relative velocity after impact. It is also referred as the ratio of kinetic energies before and after impact.

Calculate the velocity of block A just before collision using the equation of conservation of energy. Then frame the relation between the velocity of the block and velocity of the plate using the equation of coefficient of restitution. Then, frame the second equation by relating the velocity of the block with the velocity of the plate using the equation of conservation of momentum. Finally, solve both equations to obtain the velocity of the plate just after the collision.

Fundamentals

The relation for the linear momentum of a particle is given by,

Here, the mass of the particle is and the velocity of the particle is .

Write the equation of potential energy of an object.

Here, the mass of the object is , the acceleration due to gravity is, and the height at which the object is held is .

Write the formula for calculating the kinetic energy of the object.

Here, velocity at which the object moves is .

Write the equation of conservation of energy.

Here, the initial kinetic energy is , the initial potential energy is , the final kinetic energy is , and the final potential energy is .

The formula to calculate the coefficient of restitution is as follows:

Here, the velocities of two objects before the impact are and , respectively, and the velocities of the objects after the impact are and , respectively.

The schematic diagram of the system is drawn as shown in Figure (1).

Here, the height is h and the spring stiffness is k.

The initial kinetic energy of block A is zero because it is released from rest and the initial potential energy is also zero since datum is drawn at the initial position of the block.

Calculate the final kinetic energy of block A.

Substitute for v.

Calculate the final potential energy of block A.

Substitute for h.

Calculate the velocity of block A from the equation of conservation of energy.

Substitute 0 for , 0 for , for , and for .

The collision of the block A and plate B is drawn as shown in Figure (2).

Here, the velocity of the block after collision is , the velocity of the block before collision is , the velocity of the plate after collision is , and the velocity of the plate before collision is .

Write the equation of coefficient of restitution and form an equation between the final velocities of block A and plate B.

Substitute 0 for and for .

…… (1)

Write the equation of conservation of momentum and form an equation between the final velocities of block A and plate B.

Substitute for , for , for , for , for , for , for , and for

Substitute 0 for , m for , 2m for , and for .

…… (2)

Solve Equations (1) and (2).

Ans:

The velocity of the plate just after collision is .