Question

A small block of mass m1 is released from rest at the top of a curve-shaped, frictionless wedge which sits on a frictionless horizontal surface. The height of the wedge is h=5m. When the block leaves the wedge its velocity relative to the ground is measured to be 4 m/s to the right. If the mass of the block is doubled to become 2m1 what, what will its speed be when it leaves the wedge?

Answer 1

This is how far I got:
$1. \ \frac{1}{2} m_1 v_1^{2} + \frac{1}{2} m_2 v_2^{2} = m_1gh \\ \\2.\ m_1v_1 + m_2v_2 = 0 \\ \\ for \ v_1 = 4 \\ \\3.\ v_2 = -4 \frac{m_1}{m_2}$

put equation 3. into 1. together with h = 5:
4. $\frac{m_1}{m_2} = \frac{5}{8} g -1$

when I double m₁ the ratio becomes:
5. $\frac{2m_1}{m_2} = \frac{5}{4} g -2$

when I put these to ratios back into equation nr 1 and nr2, I find the ratio:

$\frac{new\ v_1}{old\ v_1} = 0.737$

and new v₁ = 2.95

I'm not sure this is correct.

Answer 2

I think "m and M" might be clearer than "m1 and m2."

There are no external horizontal forces applied, so horizontal momentum is conserved. For the ramp of mass M,
p = 0 = m*4.00m/s - M*V
giving V a positive sign. So
V = 4m/s * m / M

Since there is no friction, energy is also conserved, so the block's initial PE becomes KE in the two objects:
m*g*h = ½(m*(4.00m/s)² + M*(4m/s * m/M)²)
m * 9.8m/s² * 5m = m*8m²/s² + 8m²/s² * m²/M → divide by m
49m²/s² = 8m²/s² + 8m²/s² * m/M
m/M = 41m²/s² / 8m²/s² = 5.125
M = m / 5.125

For a block of doubled mass, our conservation of momentum equation becomes
p = 0 = 2m*v - (m/5.125)*V
which rearranges to
V = 10.25v

and so our conservation of energy equation becomes
2m*g*h = ½(2m*v² + (m/5.125)*(10.25v)²) = mv² + 10.25mv² → m cancels
2*g*h = 11.25v²
g*h = 5.625v²
so
v² = g*h / 5.625 = 49m²/s² / 5.625 = 8.71 m²/s²
v = 2.95 m/s ◄

Check:
for the block of mass m, V = 4m/s * 5.125 = 20.5 m/s
for a total KE = ½(m*4² + (m/5.125)(20.5)²) = 49m J → for m in kg

for the block of mass 2m, V = 10.25 * 2.95m/s = 30.25 m/s
and the total KE = ½(2m*(2.95)² + (m/5.125)(30.25)²) = 98m J → for m in kg

and we have in fact finished with twice the KE (which is good, since we finished with twice the starting PE).

Hope this helps!