K.E= p^2/2m.here k.E is directly proportional to p^2.given, p is doubled .then we get,k.E1/KE2= p^2/(2p)^2.k E1/kE2= p^2/4p^2.kE 1/kE2 1/4.KE2= 4 KE.( i.e it has increased 4 time than that of initial )

Momentum and Kinetic Energy

In order to understand how the kinetic energy of an object changes when its momentum is doubled, we must first understand the concepts of momentum and kinetic energy.

1. Momentum:
Momentum is a fundamental concept in physics that describes the quantity of motion of an object. It is defined as the product of an object's mass and its velocity. Mathematically, momentum (p) is given by the equation:

p = m * v

where p is momentum, m is mass, and v is velocity. Momentum is a vector quantity, meaning it has both magnitude and direction.

2. Kinetic Energy:
Kinetic energy is the energy an object possesses due to its motion. It is defined as one-half the product of an object's mass and the square of its velocity. Mathematically, kinetic energy (KE) is given by the equation:

KE = (1/2) * m * v^2

where KE is kinetic energy, m is mass, and v is velocity. Kinetic energy is a scalar quantity, meaning it has only magnitude.

Change in Kinetic Energy when Momentum is Doubled

Now, let's consider the scenario where the momentum of an object is doubled. This means that the momentum after the change is twice the momentum before the change. Mathematically, we can express this as:

p_after = 2 * p_before

Using the momentum equation (p = m * v), we can substitute the values to get:

m_after * v_after = 2 * (m_before * v_before)

Simplifying this equation, we find that:

m_after * v_after = 2 * m_before * v_before

Since we are only concerned with the change in kinetic energy, we can compare the kinetic energies before and after the change.

Comparing the kinetic energy equations (KE = (1/2) * m * v^2), we find that:

KE_after = (1/2) * m_after * v_after^2
KE_before = (1/2) * m_before * v_before^2

Substituting the values we obtained earlier for momentum, we get:

KE_after = (1/2) * (2 * m_before) * (v_before)^2

Simplifying this equation, we find that:

KE_after = 2 * (1/2) * m_before * (v_before)^2

The kinetic energy after the change is equal to twice the initial kinetic energy. Therefore, the kinetic energy is increased by a factor of 4 (2 * 2 = 4) when the momentum is doubled.

In conclusion, when the momentum of an object is doubled, its kinetic energy increases by a factor of 4. Therefore, the correct answer is option 'A' - increases 4 times.