Physics Homework, conservation of momentum and relative velocities

[wingding1]

Homework Statement

A gun, mass M , fires a bullet, mass m, with “muzzle speed” v_m (that is, vm is the speed of the
bullet relative to the gun). Ignore any other forces on the gun. Calculate the speed that the bullet
moves relative to the stationary ground, v_b. (Assume that the whole system was initially at rest.)
(b) Confirm that v_b approaches appropriate values in two limiting cases:
• M ≫ m (the “solid gold Nerf gun” limit)
• M ≪ m (the “Daffy Duck” limit)
(c) Now, imagine that a cannon, of mass M , fires a cannonball, mass m, at speed v_m at an angle θ
relative to the horizontal axis. The cannon will recoil purely horizontally with velocity v_c relative
to the ground, while the bullet will fire with horizontal velocity vb cos θ and vertical velocity
vb sin θ. Take the muzzle speed to be related to these things by:
v_m^2 = (v_b sin θ)^2 + (v_b cos θ − v_c)^2.
Calculate v_b in terms of v_m, θ, and the two masses in this case (do not take v_c to be a given
quantity.
(d) Confirm that your result from part (c) reduces to the appropriate results (from part (a)) in the
cases θ = 0 and θ = 90◦

Relevant Equations

v_m^2 = (v_b sin θ)^2 + (v_b cos θ − v_c)^2.
v_b = Mv_m/(m-+ M)

I calculated parts a) and b), and got the solution v_b = Mv_m/(m+ M) for part a (although the parts a and b are not relevant to my question), but for part c I'm confused on what to interpret v_c as. I got v_c = -v_bcosθ, so I got v_b = v_m(3cosθ+sinθ)^(-1/2), but I'm not sure if this is correct as verfiying through part d doesn't make much sense to me either, plugging in any angle doesn't give me a result with either M or m. I asked about the wording but I don't think he wrote the problem set, as he wasn't too sure either. Any ideas ?

 

[kuruman]

but for part c I'm confused on what to interpret v_c as.

As the problem states, v_c is the recoil velocity of the gun relative to the fixed Earth as required by momentum conservation in the horizontal direction.

The problem is poorly phrased. I think you need to assume that the cannon is on a horizontal frictionless surface. This means that the cannon recoils in the horizontal direction only. Strictly speaking there is recoil of the cannon + Earth system on the vertical direction, but that is negligible.

I got v_c = -v_bcosθ, so I got v_b = v_m(3cosθ+sinθ)^(-1/2),

We cannot help you sort this out unless you show us how you got this result. Please post your work.

Please note that we prefer that you post your work in $\LaTeX$ which makes the equations more legible and easier to refer to. To learn how, please click on the link "LaTeX Guide", lower left above "Attach files." Thank you.