Question

Two bodies of different mass have same kinetic energy. Compare
their momentum.

Answer 1

Given:

Two bodies of different mass have same kinetic energy

To Find:

Relation between the momentum of given bodies

Solution:

Let mass and velocity of first body be m and v respectively. Also, let the mass and velocity of second body be m' and v' respectively.

We know that,

• Expression for Kinetic Energy of a body

$\pink{\boxed{\bf{Kinetic\:Energy=\frac{1}{2}mv^{2}}}}$

• Expression for momentum of a body

$\pink{\boxed{\bf{Momentum=mv}}}$

where

m is mass of body

v is velocity of body

Now,

Let the Kinetic energy of first body be K and Kinetic energy of second body be K'

So,

$\longrightarrow\mathrm{K=\dfrac{1}{2}mv^{2}}$

Also,

$\longrightarrow\mathrm{K'=\dfrac{1}{2}m'(v')^{2}}$

According to question,

$\longrightarrow\mathrm{K=K'}$

$\longrightarrow\mathrm{\dfrac{1}{\cancel{2}}mv^{2}=\dfrac{1}{\cancel{2}}m'(v')^{2}}$

$\longrightarrow\mathrm{mv^{2}=m'(v')^{2}}$

$\longrightarrow\mathrm{\dfrac{mv}{m'v'} =\dfrac{v'}{v}}-----(1)$

Now, let the momentum of first body be M and second body be M'

So,

$\longrightarrow\mathrm{M=mv}$

Also,

$\longrightarrow\mathrm{M'=m'v'}$

Now,

$\longrightarrow\mathrm{\dfrac{M}{M'}=\dfrac{mv}{m'v'}}$

From (1),

$\longrightarrow\mathrm{\green{\dfrac{M}{M'}=\dfrac{v'}{v}}}$

which is the required relation