Question

A shot fired from a cannon explodes in the air. What will be the change in momentum and Kinetic energy?

Answer 1

The total momentum always remains conserved in an explosion because it is an isolated system and no external forces are acting on it. If the change in kinetic energy comes out to be positive then it shows that kinetic energy is increasing. So if we want to show that kinetic energy is increasing then we have to prove that change in Kinetic energy is a positive value.

Complete step-by-step answer:
When the shot is fired from the cannon then there will be no change in momentum because no external forces are applied on it .Only internal forces are acting on it . So the momentum will be conserved as it will remain zero.
A cannon explodes the shot then the internal force splits it into various parts, each having different mass and different velocities. Before an explosion the momentum is zero because the objects are at rest and so their velocities will be zero. So when we find the individual momentum of each part and then find the sum then it will be equal to the momentum before collision and hence it proves that momentum is conserved.
Let us suppose that initially the momentum is${{p}_{i}}$,
So Momentum before explosion, ${{p}_{i}}=mv$
After explosion it splits up into different objects so there momentum will be,
Momentum after explosion, ${{p}_{f}}={{m}_{1}}{{v}_{1}}+{{m}_{2}}{{v}_{2}}$
According to Law of conservation of momentum
${{p}_{i}}={{p}_{f}}$
\Rightarrow $$$$mv={{m}_{1}}{{v}_{1}}+{{m}_{2}}{{v}_{2}}
When the shot is fired from the cannon then the kinetic energy of the particle increases. Initially the kinetic energy of the particle is zero because the velocity is zero but after explosion it increases as the potential energy is changing into kinetic energy.
We know that,
Change in Kinetic energy = Final Kinetic energy –Initial Kinetic energy
After an explosion the object splits up into many particles, let us suppose it splits up into two particles having different mass and velocities and both of them are moving in different directions.
Initially mass of object $=m$
Initial velocity of object $=v$
So, Initial Kinetic energy $=\dfrac{1}{2}m{{v}^{2}}$
After explosion,
Mass of first particle becomes $=\dfrac{m}{2}$
Velocity of first particle$={{v}_{1}}$
Mass of second particle becomes$=\dfrac{m}{2}$
Velocity of second particle=$=-{{v}_{2}}$(as moving opposite to the first particle)
So Final kinetic energy becomes,

& K.E=\dfrac{1}{2}\left( \dfrac{m}{2} \right){{\left( {{v}_{1}} \right)}^{2}}+\dfrac{1}{2}\left( \dfrac{m}{2} \right){{\left( -{{v}_{2}} \right)}^{2}} \\\ & \Rightarrow K.E=\dfrac{m}{4}{{v}_{1}}^{2}+\dfrac{m}{4}{{v}_{2}}^{2} \\\ \end{aligned}$$ Now change in kinetic energy= final Kinetic energy -initial Kinetic energy $$\begin{aligned} & \Rightarrow \left( \dfrac{m}{4}{{v}_{1}}^{2}+\dfrac{m}{4}{{v}_{2}}^{2} \right)-\left( \dfrac{1}{2}m{{v}^{2}} \right) \\\ & \Rightarrow \dfrac{m}{4}\left( {{v}_{1}}^{2}+{{v}_{2}}^{2}-2v \right) \\\ \end{aligned}$$ As momentum is conserved, therefore momentum before explosion and after explosion would be same. So, $$\begin{aligned} & mv=\dfrac{m}{2}{{v}_{2}}+\dfrac{m}{2}(-{{v}_{1}}) \\\ & \Rightarrow mv=\dfrac{m}{2}\left( {{v}_{2}}-{{v}_{1}} \right) \\\ & \Rightarrow 2v={{v}_{2}}-{{v}_{1}} \\\ \end{aligned}$$ On squaring both the sides $$\begin{aligned} & {{(2v)}^{2}}={{({{v}_{2}}-{{v}_{1}})}^{2}} \\\ & \Rightarrow 4{{v}^{2}}={{({{v}_{2}})}^{2}}+{{({{v}_{1}})}^{2}}-2{{v}_{2}}{{v}_{1}} \\\ & \Rightarrow 4{{v}^{2}}+2{{v}_{2}}{{v}_{1}}={{v}_{2}}^{2}+{{v}_{1}}^{2} \\\ \end{aligned}$$ Putting this in above equation, we get Change in kinetic energy$$=\dfrac{m}{4}\left( 4{{v}^{2}}+2{{v}_{1}}{{v}_{2}}-2{{v}^{2}} \right)$$ $$\Rightarrow \dfrac{m}{4}\left( 2{{v}^{2}}+2{{v}_{2}}{{v}_{1}} \right)$$. This comes out to be a positive value which is greater than zero which proves that change in kinetic energy increases after explosion. **Note:** Momentum is of two type, angular momentum and linear momentum. An object travelling with velocity in the straight line path will have linear momentum and the spinning object will have angular momentum. Since momentum actually measures the quantity of motion possessed by the body.