Freefall motion: velocity, acceleration, momentum, kinetic energy

[SA32]

Homework Statement

Consider three ball bearings, one with a mass of 8 g, one with a mass of 16 g and one with a mass of 32 g.
a.) What heights should the three ball bearings be dropped from so that at impact all three ball bearings will have the same velocity?
b.) What heights should the three ball bearings be dropped from so that at impact all three ball bearings will have the same acceleration?
c.) What heights should the three ball bearings be dropped from so that at impact all three ball bearings will have the same momentum?
d.) What heights should the three ball bearings be dropped from so that at impact all three ball bearings will have the same kinetic energy?

Homework Equations

F=ma
v²_f=v²_i+2ad
p=mv
KE=(1/2)mv²

The Attempt at a Solution

a.) I know that in free fall motion, neglecting air resistance, objects have the same and constant acceleration, 9.81 m/s² downward. If I release all three objects from the same height at the same time, they will all hit the ground at the same time. Doesn't this mean that in order for the velocity of the three objects to be the same at impact, the height all three are dropped from must be the same? If not, I'm confused... I don't know how to relate mass and velocity in this case.
b.) Since it is free fall motion and we can neglect air resistance, does the height really matter? Aren't the balls all going to have an acceleration of 9.81 m/s² at every point in their motion?
c.) p=mv, so obviously, since the masses are different, the velocities must also be different in order for all balls to have the same momentum. Building on what I said in a.), if the velocities are different it means they were dropped from different heights.

I said m₁ = 8 g, m₂ = 16 g, and m₃ = 32 g

Then p=m₁v₁=m₂v₂=m₃v₃

We know "m" and that "p" is constant for all balls, so I tried to condense all the velocity variables into one "v" like so...

v₁=v

p=8v=16(1/2)v=32(1/4)v

But that doesn't make sense because, solving for v, I get 1 m/s... and also because, using this, I could also say that v₃ = v, then p=8(4v)=16(2v)=32v and end up with a different momentum.
d.) Since kinetic energy involves a similar relationship to momentum I think I can try this again with some tips on how to approach c.)

Thank you!

 

[AlephZero]

a.) I know that in free fall motion, neglecting air resistance, objects have the same and constant acceleration, 9.81 m/s² downward. If I release all three objects from the same height at the same time, they will all hit the ground at the same time. Doesn't this mean that in order for the velocity of the three objects to be the same at impact, the height all three are dropped from must be the same?

Correct.

b.) Since it is free fall motion and we can neglect air resistance, does the height really matter? Aren't the balls all going to have an acceleration of 9.81 m/s² at every point in their motion?

Correct.

c.) ...We know "m" and that "p" is constant for all balls, so I tried to condense all the velocity variables into one "v" like so...

v₁=v

p=8v=16(1/2)v=32(1/4)v

But that doesn't make sense...

It doesn't make sense because the velocities are not the same!

What you mean is p = 8v₁ = 16v₂

Therefore v₂ = 1/2 v₁.

NB you are not given the actual value of the momentum, so you can only find the relative heights (i.e find h2 and h3 in terms of h1)

 

[SA32]

Thanks so much! Now I think I can use the kinematics equation to find the relations between the heights, and do likewise for the kinetic energy question. Cool.