Work Energy And Power

The energy equivalent of a substance can be found using Einstein's famous equation: where:
is the energy,
is the mass of the substance,
is the speed of light ( ).

Given:
.
Now, using the formula: Thus, the energy equivalent of 5 g of the substance is .

The correct option is (C) :

We know that the kinetic energy of a body is given by the formula: Where: - is the mass, - is the velocity of the body. For the two bodies, the kinetic energies are in the ratio 1 : 2, so: Substitute the kinetic energy formula: Simplifying: The momentum of a body is given by: Thus, the ratio of the momenta and is:

Thus, the momenta are in the ratio .

Given:

• Particle KE: .
• Rolling sphere KE: Translational + Rotational = .
• For a solid sphere: and .

Step 1: Equate Kinetic Energies

Simplify:

Conclusion:

The mass ratio is .

Given:

• An object at rest explodes into three equal masses ( ).
• Two parts move at right angles with speed each.

Step 1: Momentum Conservation

Initial momentum = 0 (object at rest).

Let the third part have velocity . Momentum conservation requires:

Given and , the third momentum must satisfy:

Thus, the speed of the third part is:

The speed of the third part is .

Given:

• Ball moves with velocity , collides with stationary .
• After collision, deflects by , and the angle between and is .

Step 1: Apply Momentum Conservation

Let and be the speeds of and after collision.

Horizontal:

Vertical:

Step 2: Solve for

From vertical momentum: .

Substitute into horizontal momentum:

Conclusion:

The speed of after collision is .

Given:

• Mass of each bullet,
• Speed of each bullet,
• Power delivered by gun,
• Number of bullets fired per minute =

Step 1: Calculate Kinetic Energy per Bullet

The kinetic energy (KE) of one bullet is:

Step 2: Determine Total Energy per Minute

Power is energy per unit time. For bullets per minute (60 seconds):

Rearrange to solve for :

Conclusion:

The number of bullets fired per minute, , is 300.

Answer:

Given:

• Body moving under constant power (P = constant)
• Displacement at t = 0 s: 0 m
• Displacement at t = 10 s: 10 m

Step 1: Relate Power to Motion

For constant power, the work done (W) is proportional to time (t):

Work translates to kinetic energy: . Thus, velocity (v) depends on time as:

Step 2: Integrate Velocity to Find Displacement

Displacement (s) is the integral of velocity:

Thus, displacement follows:

Step 3: Determine Proportionality Constant (k)

Given :

Step 4: Calculate Displacement at t = 20 s

Conclusion:

The displacement at is .

Answer:

Given:

• Frictional force,
• Stopping distance,
• Initial speed,
• Final speed, (object comes to rest)

Step 1: Calculate Deceleration

Using the kinematic equation:

Substitute known values:

(Negative sign indicates deceleration)

Step 2: Relate Force and Mass

Using Newton's Second Law:

Conclusion:

The mass of the object is 10 kg.

Answer: