Derive the unit of force using the second law of motion. $A$ force of $5 \, N$ produces an acceleration of $8 \, m s^{-2}$ on a mass $m_1$ and an acceleration of $24 \, m s^{-2}$ on a mass $m_2$. What acceleration would the same force provide if both the masses are tied together (in $, m s^{-2}$)?

It is required to increase the velocity of a scooter of mass $80 \ kg$ from $5 \ m s^{-1}$ to $25 \ m s^{-1}$ in $2 \ s$. Calculate the force required.

Two objects of masses $100 \, g$ and $200 \, g$ are moving along the same line and direction with velocities of $2 \, m \, s^{-1}$ and $1 \, m \, s^{-1}$,respectively. They collide and after the collision,the first object moves at a velocity of $1.67 \, m \, s^{-1}$. Determine the velocity (in $m/s$) of the second object.

If the time taken to bring a ball to rest from a certain velocity $v$ is reduced to half,what will be the changes in values of
$(a)$ Initial and final momentum.
$(b)$ Change of momentum.
$(c)$ Rate of change of momentum.

$A$ bullet of mass $10\, g$ travelling horizontally with a velocity of $150\, m/s$ strikes a stationary wooden block and comes to rest in $0.03\, s$. Calculate the distance of penetration of the bullet into the block. Also,calculate the magnitude of the force exerted by the wooden block on the bullet.

Calculate the ratio of momentum,when
$(i)$ velocity of an object is doubled.
$(ii)$ mass of an object is halved.
$(iii)$ both mass and velocity are increased by three times.