Question

If the kinetic energy of a body becomes four times of its initial value, then the new momentum
1. will be three times of the initial value
2. will be four times of the initial value
3. will be two times of the initial value
4. will remain unchanged

Answer 1

Correct Answer - Option 3 : will be two times of the initial value

Concept:

Momentum

• Momentum is defined as the product of the mass and velocity of the body. 
• It is a vector quantity directed toward velocity. 
• It is given as 

p = mv -- (1)

m is mass, v is the velocity, p is momentum

Kinetic Energy

• The energy of a body in motion due to its state of motion is called kinetic energy. 
• It is given as 

\(K = \frac{1}{2}mv^2\) --- (2)

K is kinetic energy, m is mass, v is speed.

Relationship between Momentum and Kinetic Energy

If we combine equation (1) and (2) we will get the relationship between momentum and kinetic energy as

\(K = \frac{p^2}{2m}\) -- (3)

Calculation:

So, let the initial kinetic energy of the body be K, momentum be p, and let its mass be m

Then from Equation (3)

\(K = \frac{p^2}{2m}\)

Or

p² = 2mK -- (4)

Now, It is given Kinetic energy becomes 4 times of its initial value. Then, we can say that new kinetic energy is K' (K' = 4K) and new momentum is p'.

\(K '= \frac{p'^2}{2m}\)

or

p'² = 2mK'

⇒ p'² = 2m (4K)

⇒ p'2 = 4 (2m K)

⇒ p'2 = 4 p (From Equation 4)

Taking Square root of both side

⇒ p' ² = 2 p

So, p' the final momentum is twice of initial momentum p. 

Therefore, the correct option is will be two times of the initial value.