How Is Kinetic Energy Distributed in an Explosion with Unequal Mass Fragments?

[ArmandoC]

Homework Statement

An explosion breaks an object into two pieces, one of which has 1.5 times the mass of the other. If 7400 J were released in the explosion, how much kinetic energy did each piece acquire?

Homework Equations

Momentum?
Kinetic Energy?

The Attempt at a Solution

I used all types of formulas having to do with energies and momentum and nothing gave me an answer.

 

[chaoseverlasting]

Use conservation of linear momentum along the horizontal since all the forces are internal (the energy that the explosion released was stored inside the projectile/object).

You know that m1v1+m2v2=0. And are given that m1=1.5m2.
Also, the total energy released in the explosion is converted into kinetic energy (you have to assume that), so $$0.5m_1v_1^2+ 0.5m_2v_2^2=7500$$. Solve for v1/v2 and find the KE of each piece.

 

[m2003]

I would approach this problem by first considering the conservation of energy and momentum. In an explosion, the total energy before and after the explosion must be the same. Additionally, the total momentum before and after the explosion must also be the same.

Using the given information, we can set up the following equations:

1) Conservation of energy: 7400 J = KE1 + KE2, where KE1 and KE2 represent the kinetic energies of the two pieces after the explosion.

2) Conservation of momentum: m1v1 + m2v2 = 0, where m1 and m2 represent the masses of the two pieces and v1 and v2 represent their respective velocities.

Since we know that one piece has 1.5 times the mass of the other, we can represent this relationship as m2 = 1.5m1.

Using this relationship, we can solve for m1 in terms of m2 in equation 2:

m1v1 + 1.5m1v2 = 0

m1(v1 + 1.5v2) = 0

m1 = -1.5m2

Now, we can substitute this value for m1 into equation 1:

7400 J = KE1 + KE2

7400 J = 0.5m1v1^2 + 0.5m2v2^2

7400 J = 0.5(-1.5m2)v1^2 + 0.5m2v2^2

7400 J = -0.75m2v1^2 + 0.5m2v2^2

We can then rearrange this equation to solve for v1^2 in terms of v2^2:

v1^2 = (1.5/0.75)v2^2 - (7400/0.75)

v1^2 = 2v2^2 - 9866.67

Now, we can plug this value for v1^2 into equation 2 and solve for v2:

m1v1 + m2v2 = 0

-1.5m2v2 + m2v2 = 0

v2 = 0 m/s

Since v2 = 0, we can then solve for v1:

v1^2 = 2(0)^2