How to find combined velocity after impact in a mallet and stake collision?

[Depasquale97]

Homework Statement

A heavy mallet of Mass M (see Attached file) is dropped and moves through vertical distance y before it hits the top of a tent stake of mass m, driving it into the ground a distance d. Assume that the ground offers a a constant resistance to the motion of the tent stake and mallet, which move together after impact.
Show that the combined velocity, v_c, of the mallet and the tent stake after impact in terms of the mass of the mallet M, the mass of the stake m and the velocity of the mallet before impact v is given by
v_c=v/(1+m/M)

Homework Equations

The Attempt at a Solution

I tried to use the equation; Total momentum before collision = Total momentum after collision but had no success.

 

[RUber]

You know that at the point of impact the momentum of the mallet is Mv and that momentum is transferred to the combined system as (M+m)v_c. Is any momentum lost?
If not, do the algebra.

 

[Depasquale97]

Thank you RUber, no momentum is lost.

 

[RUber]

So then you need to show that
$Mv = (M+m)v_c \iff v_c = \frac{v}{1+\frac{m}{M}}$
Do this by isolating v_c and rewriting (M+m) as M(1+m/M).

 

[Depasquale97]

Thank you very much, much appreciated