Question 42: Work, Energy and Power & Circular Motion

Question

When kinetic energy of a body becomes 36 times of its original value, the percentage increase in the momentum of the body will be :

Accepted Answer

The relationship between kinetic energy (K.E) and momentum (p) of a body can be expressed through their respective definitions. Kinetic energy is given by $$K.E = \frac{1}{2} mv^2$$ where $m$ is the mass of the body and $v$ is its velocity. The momentum (p) of a body is given by $$p = mv$$. To express kinetic energy in terms of momentum, we can manipulate the expression for momentum as follows:
$$p = mv \implies v = \frac{p}{m}$$
Substituting $v$ in the kinetic energy formula, we get
$$K.E = \frac{1}{2} m\left(\frac{p}{m}ight)^2 = \frac{1}{2} \frac{p^2}{m}$$
Therefore, we see that kinetic energy is directly proportional to the square of the momentum $(K.E \propto p^2)$.
Now, given that the kinetic energy of a body becomes 36 times its original value, we can set up the proportionality as
$$\frac{K.E_{	ext{final}}}{K.E_{	ext{original}}} = 36$$
Since $K.E_{	ext{final}} = 36 	imes K.E_{	ext{original}}$ and knowing $K.E \propto p^2$, we can express this relationship through the squares of the initial and final momentum:
$$\frac{p_{	ext{final}}^2}{p_{	ext{original}}^2} = 36$$
Taking the square root of both sides to find the ratio of final to initial momentum, we have
$$\frac{p_{	ext{final}}}{p_{	ext{original}}} = \sqrt{36} = 6$$
This indicates that the final momentum is 6 times the original momentum. To find the percentage increase in the momentum, we calculate the increase from the original to the final, subtracting the original momentum (which is considered 1 times itself):
$$	ext{Percentage increase} = \left(\frac{p_{	ext{final}} - p_{	ext{original}}}{p_{	ext{original}}}ight) 	imes 100\% = \left(\frac{6p - p}{p}ight) 	imes 100\% 
= \left(6 - 1ight) 	imes 100\% = 5 	imes 100\% = 500\%$$
Therefore, the correct answer is Option B: 500%.

Answer

60%

Answer

500%

Answer

6%

Answer

600%

Work, Energy and Power & Circular Motion

When kinetic energy of a body becomes 36 times of its original value, the percentage increase in the momentum of the body will be :

Choose the correct answer: