If a man jumps out from a boat to the bank of the river,the boat moves backwards. Identify the action and reaction in this situation.

$(a)$ State Newton's second law of motion and show that the first law of motion can be mathematically derived from the mathematical expression for the second law of motion.
$(b)$ A stone dropped from a window reaches the ground in $0.5 \, s$ (given $g = 10 \, m \, s^{-2}$):
$(i)$ Calculate the speed just before it hits the ground.
$(ii)$ What is the average speed during $0.5 \, s$?
$(iii)$ Calculate the height of the window from the ground.

Name and state the action and reaction in the following cases:
$(i)$ Firing of a bullet from a gun.
$(ii)$ Hammering a nail.
$(iii)$ $A$ book lying on a table.
$(iv)$ Moving rocket.
$(v)$ $A$ person moving on the floor.
$(vi)$ $A$ moving train colliding with a stationary train.

In oil tankers,some space is left at the top while filling them. Explain why.

State Newton's third law of motion.

$A$ man weighing $60 \, kg$ runs along the rails with a velocity of $18 \, km \, h^{-1}$ and jumps into a car of mass $1 \, \text{quintal} \, (100 \, kg)$ standing on the rails. Calculate the velocity with which the car will start travelling along the rails.