Calculating Rest Mass of Original Object in Inelastic Explosion

[ftbf22]

Homework Statement

An object at rest in deep space explodes into two parts converting some of its
original mass into kinetic energy of the fragments. Both fragments move at a speed of
0.60 c relative to the position of the original object. Each fragment has a rest mass of
3.0 kg. Calculate the rest mass of the original object.

Homework Equations

Conservation of momentum m₀v₀ = m₁v₁ + m₂v₂

Mass-energy relationship E=mc^2

The Attempt at a Solution

Kinetic energy is not conserved since explosion is inelastic. This is when I get stuck: how can I represent the kinetic energy ('lost mass') in the form of momentum?

 

[haruspex]

Momentum will not help you here. The two fragments will have equal and opposite momentum, and there's nothing more to be deduced.
I'm not sure what you mean by describing the explosion as 'inelastic'. Yes, of course there would be losses in practice, but it seems to me that the question expects you to ignore that..

 

[rude man]

orig rest energy = post-rest energy + kinetic energy of the two fragments, as seen from the position of the explosion.

 

[tiny-tim]

welcome to pf!

hi ftbf22! welcome to pf!

Kinetic energy is not conserved since explosion is inelastic.

total energy is conserved, so just add up the total energies

This is when I get stuck: how can I represent the kinetic energy ('lost mass') in the form of momentum?

what does momentum have to do with it?

 

[Nickie]

I would approach this problem by first defining the variables and equations that are relevant to the situation. In this case, we have an explosion that converts some of the original mass into kinetic energy of the fragments. We also know the rest mass and speed of the fragments, as well as the rest mass of the original object. With this information, we can use the principles of conservation of momentum and mass-energy equivalence to solve for the rest mass of the original object.

First, let's define our variables:

m0 = rest mass of original object
m1 = rest mass of fragment 1
m2 = rest mass of fragment 2
v1 = speed of fragment 1
v2 = speed of fragment 2

Now, let's use the conservation of momentum equation:

m0v0 = m1v1 + m2v2

Since the fragments are moving at the same speed (0.60 c), we can simplify this equation to:

m0v0 = (m1 + m2)v1

We also know that the rest mass of each fragment (m1 and m2) is 3.0 kg. Substituting this into the equation, we get:

m0v0 = (3.0 kg + 3.0 kg)(0.60 c)

Next, we need to convert the speed from c to m/s. We can do this by multiplying by the speed of light, c, which is approximately 3.0 x 10^8 m/s.

m0v0 = (3.0 kg + 3.0 kg)(0.60 x 3.0 x 10^8 m/s)

Simplifying, we get:

m0v0 = 3.6 x 10^8 kg m/s

Now, we can use the mass-energy relationship, E=mc^2, to relate the kinetic energy (lost mass) to the momentum:

E = m0c^2 = (3.6 x 10^8 kg m/s)^2/c^2

Solving for m0, we get:

m0 = (3.6 x 10^8 kg m/s)^2/c^4

Plugging in the value for c, we get:

m0 = (3.6 x 10^8 kg m/s)^2/ (3.0 x 10^8 m/s)^2