A 41.0-kg body is moving in the direction of the positive x axis with a speed of 306 m/s when, owing to an internal explosion, it breaks into three pieces. One part, whose mass is 8.0 kg, moves away from the point of explosion with a speed of 326 m/s along the positive y axis. A second fragment, whose mass is 4.5 kg, moves away from the point of explosion with a speed of 431 m/s along the negative x axis. What is the speed of the third fragment? Ignore effects due to gravity.

Question

A 41.0-kg body is moving in the direction of the positive x axis with a speed of 306 m/s when, owing to an internal explosion, it breaks into three pieces. One part, whose mass is 8.0 kg, moves away from the point of explosion with a speed of 326 m/s along the positive y axis. A second fragment, whose mass is 4.5 kg, moves away from the point of explosion with a speed of 431 m/s along the negative x axis. What is the speed of the third fragment? Ignore effects due to gravity.

How much energy was released in the explosion?

Answer 1

To find the speed of the third fragment, we can first calculate the total momentum before and after the explosion, and then use the conservation of momentum principle.

Before the explosion:
The momentum of the 41.0 kg body moving in the positive x direction is given by:
p1 = m1 * v1

Given:
m1 = 41.0 kg
v1 = 306 m/s

p1 = 41.0 kg * 306 m/s

After the explosion:
Let's assume the third fragment has a mass of m3 and moves with a speed of v3.

The momentum of the first fragment moving in the positive y direction is given by:
p2 = m2 * v2

Given:
m2 = 8.0 kg
v2 = 326 m/s

p2 = 8.0 kg * 326 m/s

The momentum of the second fragment moving in the negative x direction is given by:
p3 = m3 * v3

Given:
m3 = unknown
v3 = unknown

p3 = m3 * v3

Conservation of momentum:
The total momentum before the explosion is equal to the total momentum after the explosion.

p1 = p2 + p3

So, we can write:
41.0 kg * 306 m/s = 8.0 kg * 326 m/s + m3 * v3

Now, to find the speed of the third fragment, we need one more equation.

To find the energy released in the explosion, we can use the principle of conservation of total mechanical energy.

Before the explosion, the body has kinetic energy due to its motion:
KE1 = (1/2) * m1 * v1²

After the explosion, the total mechanical energy is the sum of the kinetic energies of the fragments.
KE2 = (1/2) * m2 * v2² + (1/2) * m3 * v3²

To find the energy released, we can calculate the change in total mechanical energy:
E = KE2 - KE1

Substituting the given values, we can solve these equations simultaneously to find the speed of the third fragment and the energy released in the explosion.