Question

A cannonball (mass $ = 2{\text{ kg}}$) is fired at a target of $500{\text{ m}}$ away. The $400{\text{ kg}}$ cannon recoils by $50{\text{ cm}}$. Assuming the cannon ball moves with constant velocity, when will it hit the target? (resistance offered by ground to cannon is $100{\text{ N}}$).

Answer 1

Hint: We are given the mass of the cannon and the cannonball and the distance the cannonball must travel to hit the target. We are required to find the time and for that, we need to find the velocity of the cannonball with which it was fired and we need to make sure the laws of conservation are taken into consideration while solving it.

Complete step by step answer:
Since no external force is applied to the system, therefore, the momentum of the system is conserved and hence we get
$mv = MV$
Where $m$ is the mass of the cannonball and $v$ is its velocity and $M$ is the mass of the cannon and $V$ is its velocity.
$ \Rightarrow 2 \times v = 400 \times V$
Therefore we need to find the velocity of the cannonball.Now it is given that the ground exerts a force on the cannon. Therefore work done by the ground on the cannon is given by
$W = F \times s$
Where $s$ is the distance of the recoil of the cannon and $F$is the resistive force.
Therefore substituting the values we get
$ \Rightarrow W = 100 \times 0.5{\text{ J}}$
$ \Rightarrow W = 50{\text{ J}}$

Now applying the law of conservation of kinetic energy between the ground and cannon we get work done by the ground on the cannon is equal to the kinetic energy of the cannon immediately after firing.
$50{\text{J}} = \dfrac{1}{2}M{V^2}$
Substituting the values we get
$V = 0.5{\text{ m}}{{\text{s}}^{ - 1}}$
Now substituting the value of the velocity of the cannon in the conservation of momentum equation we get,
$ \Rightarrow 2 \times v = 400 \times 0.5$
$ \therefore v = 100{\text{ m}}{{\text{s}}^{ - 1}}$
Therefore time is taken by the cannonball to hit the target $ = \dfrac{{500}}{{100}} = 5\sec $.

Note: When no external force is applied to an object, the momentum of the object does not change which means it remains conserved even when the cannonball is fired. The kinetic energy of the cannon after firing gets converted into the work done by the ground which creates a recoil displacement in the cannon which is the law of conservation of energy.