Explosion and conservation of momentum problem

[Pochen Liu]

Note: Please only give hints please! No answers because I want the satisfaction of solving it.

1. Homework Statement
A mass M at height h above flat round and falling vertically with velocity v breaks up explosively into 2 parts. The kinetic energy given to the system in the explosion is E. The 2 halves leave the point of explosions at angles theta and psi with respect to the downward vertical.

State the conservation laws that apply to the motion of the exploding system and neglecting any change in mass, write down the equations derived by application of these laws.

Consider the case that E = mv^2 and one of the halves emerges at theta = 90 degrees. Show that cos(psi) = 2/sqrt(5).

Homework Equations

How do I prove this?

The Attempt at a Solution

I have calculated the initial and final momentum of x and y-axis but I don't know how to prove it or utilise the kinetic energy part because I'm assuming it's part of the proof.

I don't know what way to prove it, and I don't know where the numerical aspect came from because this question is almost formula based.
https://scontent.fwlg1-1.fna.fbcdn.net/v/t35.0-12/17373213_1049612521848826_812759978_o.jpg?oh=f29a20165ddd2d2f671c3f1d032e0744&oe=58CF94AC

 

[ehild]

Note: Please only give hints please! No answers because I want the satisfaction of solving it.

1. Homework Statement
A mass M at height h above flat round and falling vertically with velocity v breaks up explosively into 2 parts. The kinetic energy given to the system in the explosion is E. The 2 halves leave the point of explosions at angles theta and psi with respect to the downward vertical.

State the conservation laws that apply to the motion of the exploding system and neglecting any change in mass, write down the equations derived by application of these laws.

Consider the case that E = mv^2 and one of the halves emerges at theta = 90 degrees. Show that cos(theta) = 2/sqrt(5).

Homework Equations

How do I prove this?

The Attempt at a Solution

I have calculated the initial and final momentum of x and y-axis but I don't know how to prove it or utilise the kinetic energy part because I'm assuming it's part of the proof.

I don't know what way to prove it, and I don't know where the numerical aspect came from because this question is almost formula based.
https://scontent.fwlg1-1.fna.fbcdn.net/v/t35.0-12/17373213_1049612521848826_812759978_o.jpg?oh=f29a20165ddd2d2f671c3f1d032e0744&oe=58CF94AC

You have to state the conservation laws, valid for the system. If you say the momentum is conserved you are right, and it means that both components (horizontal and vertical) are conserved. What is the condition that the momentum conserves in a process?
What equation did you get in terms of θ and Ψ? Knowing that θ=90°, you should derive cos(Ψ) (not cos (θ) as you wrote).

The energy is not conserved in an explosion. The problem says it increases by E. Given also an angle, you have to derive a formula showing where the pieces hit the ground. Calculate the initial velocities just after the explosion. What kind of motion do the pieces perform?

 

[Pochen Liu]

You have to state the conservation laws, valid for the system. If you say the momentum is conserved you are right, and it means that both components (horizontal and vertical) are conserved. What is the condition that the momentum conserves in a process?
What equation did you get in terms of θ and Ψ? Knowing that θ=90°, you should derive cos(Ψ) (not cos (θ) as you wrote).

The energy is not conserved in an explosion. The problem says it increases by E. Given also an angle, you have to derive a formula showing where the pieces hit the ground. Calculate the initial velocities just after the explosion. What kind of motion do the pieces perform?

I have derived the equation:
total momentum (-mv) = -m₁v₁ + 1/2(m₂v₂Sin(psi)) + 1/2(m₂v₂Cos(psi))

So if the system increases by E, do I convert the equation I have for the total momentum into kinetic energy and then add E?
And because theta is a 90 degree angle how can I use trig to put in an equation if that is the right angle and the angle I'm given because I can't determine which other sides are adjacent or opposite?
And what do you mean by "what is the condition that the momentum conserves in a process?"

 

[ehild]

I have derived the equation:
total momentum (-mv) = -m₁v₁ + 1/2(m₂v₂Sin(psi)) + 1/2(m₂v₂Cos(psi))

You have equations both for the vertical and horizontal components of the momenta. What are they? See picture.

So if the system increases by E, do I convert the equation I have for the total momentum into kinetic energy and then add E?

You need to get the momenta for both pieces, and determine the individual kinetic energies. Add them, and add E.

And what do you mean by "what is the condition that the momentum conserves in a process?"

What does Newton firs law state in terms of the momentum? Is the momentum conserved if an external force acts?

 

[Pochen Liu]

You have equations both for the vertical and horizontal components of the momenta. What are they? See picture.
View attachment 114683

You need to get the momenta for both pieces, and determine the individual kinetic energies. Add them, and add E.

What does Newton firs law state in terms of the momentum? Is the momentum conserved if an external force acts?

Thank you! I've solved it :)

 

[ehild]

Thank you! I've solved it :)

Congratulation!