Why isn't (1/2)(mv₀)² = 1/2(M+m)gh valid for conservation of energy?

[j04015]

Homework Statement

Check image

Relevant Equations

Conservation of momentum and energy

Why is (1/2)(mv0)^2 = 1/2(M+m0)gh not a valid equation for conservation of energy?

Isn't the energy from when the dart is shot the same as when the two masses move at speed v?

 

[PeroK]

Homework Statement: Check image
Relevant Equations: Conservation of momentum and energy

View attachment 336233View attachment 336234
Why is (1/2)mv0 = 1/2(M+m0)gh not a valid equation for conservation of energy?

Isn't the energy from when the dart is shot the same as when the two masses move at speed v?

Your question is whether the collision between the dart and block is elastic or not?

 

[kuruman]

To answer the first question,

Why is (1/2)mv0 = 1/2(M+m0)gh not a valid equation for conservation of energy?

Because $\frac{1}{2}mv_0$ has dimensions of momentum and not energy.

 

[j04015]

To answer the first question,

Because $\frac{1}{2}mv_0$ has dimensions of momentum and not energy.

Whoops, typo. I meant (1/2)(mv0)^2

 

[j04015]

Your question is whether the collision between the dart and block is elastic or not?

Your question is whether the collision between the dart and block is elastic or not?

If the collision wasn't elastic the entire problem doesn't make sense.

 

[PeroK]

If the collision wasn't elastic the entire problem doesn't make sense.

That statement is false!

 

[PeroK]

... the collision is manifestly and totally inelastic!

 

[j04015]

... the collision is manifestly and totally inelastic!

I see the issue now. Momentum is conserved but not energy. Thanks!

 

[kuruman]

I see the issue now. Momentum is conserved but not energy. Thanks!

You can see that if you consider what's going on in the center of mass frame. Before the collision, both dart and block move with opposite momenta. Total momentum is zero and the kinetic energy is non-zero. After the collision, the dart and the block are at rest. Total momentum is zero (conserved) and total kinetic energy is also zero (not conserved).

 

[Orodruin]

You can see that if you consider what's going on in the center of mass frame. Before the collision, both dart and block move with opposite momenta. Total momentum is zero and the kinetic energy is non-zero. After the collision, the dart and the block are at rest. Total momentum is zero (conserved) and total kinetic energy is also zero (not conserved).

The conservation of energy equations will not be compatible with conservation of energy in any frame if you assume that the dart sticks. However, I agree that considering the com frame makes it very explicit.