Question

A man of $50\,\text{kg}$ is standing at one end of a boat of length $25\ \text{m}$ and mass $200\text{ kg}$. If he starts running and when he reaches the other end, he has a velocity $2m{{s}^{-1}}$ with respect to the boat. The final velocity of the boat is $\left( \text{in m}{{\text{s}}^{-1}} \right)$
(a) $\dfrac{2}{5}$
(b) $\dfrac{2}{3}$
(c) $\dfrac{8}{5}$
(d) $\dfrac{8}{3}$

Answer 1

Hint : by using conservation of momentum we can find final velocity of the boat but first we need to make a formula for final total momentum of the man and boat system.
Applying this concept we are going to calculate values of questions.
Formula used.
$P=mv$

Complete solution:
As per data given in the question we have,
(i) mass of the man ${{m}_{1}}=50\text{ kg}\text{.}$
(ii) Mass of the boat ${{m}_{2}}=200\text{ kg}$
(iii) Velocity of the man after reaching at the end ${{V}_{1}}=2\text{m/s}$
(iv) Velocity of the boat after man reaching at the end be ${{V}_{2}}$
As we know at starting man and boat where are rest.
So, initial momentum of men and boat is ${{P}_{1}}=0\ \text{kgm/s}$ and final momentum of man and boat is ${{P}_{2}}$
Now using conservation of momentum
${{P}_{2}}={{P}_{1}}$
But we know that $P=mv$ so,
${{P}_{1}}={{m}_{1}}{{v}_{1}}$
${{P}_{2}}={{m}_{1}}{{v}_{1}}+\left( {{m}_{1}}+{{m}_{2}} \right){{v}_{2}}$
Now, we get,
${{P}_{2}}={{P}_{1}}$
${{m}_{1}}{{v}_{1}}+\left( {{m}_{1}}+{{m}_{2}} \right){{v}_{2}}=0$ $\left( \therefore {{P}_{1}}=0 \right)$
Substituting all values in the above equation.
$\left( 50\times 2 \right)\left( 50+200 \right){{v}_{2}}=0$
$100+250{{v}_{2}}=0$
${{v}_{2}}=\dfrac{-100}{250}$
${{v}_{2}}=\dfrac{-2}{5}\text{ m/}{{\text{s}}^{-1}}$Negative sign shows that velocity of boat is in opposite direction to that of the man.

Hence option (a) is the correct answer.

Note:
(a) velocity is a ratio of displacement to change in time. Distance and displacement are the main key point of velocity. There will be no velocity if there is no displacement in the object or if the object does not have any distance.
There are two kinds of velocity.
(i) initial velocity: velocity of body at beginning of the given is called initial velocity.
(ii) Final velocity: velocity of body at the end of time given is called final velocity.
Mass is the amount of matter in the body. Mass does not change with time. It only changes in extreme cases like a huge amount of energy is given or taken from a body. Mass and weight are different things in physics SI unit of mass in kilogram (kg)
Momentum is called as quantity of motion of the body momentum is mass time velocity
$P=mv$
SI unit of momentum is $\text{kgm}{{\text{s}}^{\text{-1}}}$
Conservation of momentum is the law of physics; it states that when no external force is applied on the system then momentum of a system is constant.
Conservation of momentum law is embodied in the best example of conservation of momentum is space travel. In space there is no force. So rockets eject matter at very high speed and momentum remains constant. Therefore rockets move in the opposite direction of the exhaust.