Question

A mortar fires a shell of mass m at speed v0. The shell explodes at the top of its trajectory (shown by a star in the figure) as designed. However, rather than creating a shower of colored flares, it breaks into just two pieces, a smaller piece of mass 15m and a larger piece of mass 45m. Both pieces land at exactly the same time. The smaller piece lands perilously close to the mortar (at a distance of zero from the mortar). The larger piece lands a distance d from the mortar. If there had been no explosion, the shell would have landed a distance r from the mortar. Assume that air resistance and the mass of the shell's explosive charge are negligible. (Figure 1)

Find The distance d from the mortar at which the larger piece of the shell lands. Solve d in terms of r.

Answer 1

Concepts and reason

The question is based on the conservation of momentum.

Initially, find the linear momentum of two fragments. After that equate the sum of two masses to the momentum before explosion and solve for the distance of the larger piece from the mortar.

Fundamentals

Consider a particle of mass m exploding into two masses and such that .

The momentum of an object of mass m moving with velocity is given as follows:

$p=mv. $

Here, p is the momentum of the object.

The velocity of an object after travelling distance d in time t is given as follows:

Here, v is the velocity of the object.

Find the expression for the larger fragment of the object.

The momentum of smaller mass is given as follows:

Here, r is the position of center of mass from the mortar, t is the time taken by the fragments to land after the explosion and is the velocity of the smaller fragment.

The negative sign indicated that the smaller fragment moves backwards after the explosion.

The momentum of larger mass is given as follows:

Here, is the velocity of the smaller fragment and d is the distance of the larger fragment from the mortar.

The momentum of the object before explosion is,

$p=m(v.cos) $

Here, is the angle made by initial velocity with horizontal.

The horizontal component of velocity is,

$v.coso=2 $

So, the linear momentum of the shell is,

As no external force is acting on the system, so the momentum of the system remains conserved before and after the explosion.

Apply law of conservation of momentum,

Substitute for and for as follows:

Ans:

The distance d from the mortar at which larger piece of the shell lands is .