Question

Two masses of 1kg and 4 kg are moving with equal momentum the ratio of their kinetic energies

Answer 1

their kinetic energy ratio is 4:1

Answer 2

Given:

Two objects of mass $1kg$ and $4kg$.

Both the objects have equal momentum.

To find:

The ratio of the kinetic energies of the two masses.

Explanation:

The kinetic energy is the energy possessed by a body by virtue of the velocity of the object during the motion.

Mathematically,  $KE=\frac{1}{2}mv^{2}$

Multiplying numerator and denominator by mass $m$, we get

$KE=\frac{1}{2}mv^{2} (\frac{m}{m} )$

$KE=\frac{1}{2}\frac{(mv)^{2} }{m}$

We know, momentum of a moving body is given by $P=mv$ . We get

$K=\frac{P^{2} }{2m}$    $;$ $or$

$P=\sqrt{2mK}$

Solution:

According to the question, two masses lets say $m_{1} =1kg$ and $m_{2} =4kg$  . and the momentum of both the bodies are equal.

We have,

Momentum of mass $m_{1}$, $P_{1} =\sqrt{2m_{1}K_{1} }$

Momentum of mass $m_{2}$,  $P_{2}=\sqrt{2m_{2}K_{2} }$  

Hence, according to the question,

$P_{1} =P_{2}$

$\sqrt{2m_{1} K_{1} }= \sqrt{2m_{2} K_{2} }$

$m_{1} K_{1}= m_{2} K_{2}$

$\frac{K_{1} }{K_{2} }= \frac{m_{2} }{m_{1} }$

We get,

$\frac{K_{1} }{K_{2} } =\frac{4}{1}$

Final answer:

Hence, the ratio of the kinetic energy of the two objects are 4 : 1.