Using conservation of energy:
[tex]\begin{gathered} E1=E2 \\ mgh1=\frac{1}{2}mv2^2+mgh2 \end{gathered}[/tex]Solve for v2:
[tex]\begin{gathered} 2gh1=v2^2+2gh2 \\ v2=\sqrt{2g(h1-h2)} \\ v2=\sqrt{2(9.8)(6.7-6.7/2)} \\ v2\approx8.1m/s \end{gathered}[/tex]When you are half way down, your velocity is approximately 8.1 m/s
Using conservation of momentum:
[tex]m1va+m2vb=v(m1+m2)[/tex]Where:
m1 = Your mass = 62kg
m2 = your friend's mass = 71kg
va = Your velocity = 8.1m/s
vb = Your friend's velocity = 8.5m/s
v = Velocity after the collision
so:
[tex]\begin{gathered} v=\frac{m1va+m2vb}{m1+m2} \\ v=\frac{62\cdot8.1+71\cdot8.5}{62+71} \\ v=\frac{502.2+603.5}{133} \\ so: \\ v=\frac{1105.7}{133} \\ v\approx8.3m/s \end{gathered}[/tex]After the collision you and your friend are moving at 8.3 m/s
Finally, using conservation of energy again:
[tex]\begin{gathered} E2=E3 \\ \frac{1}{2}(m1+m2)v^2+(m1+m2)gh2=\frac{1}{2}(m1+m2)vf^2 \end{gathered}[/tex]Where:
vf = velocity at the bottom
so:
[tex]v^2+2gh2=vf^2[/tex]Solve for vf:
[tex]\begin{gathered} vf=\sqrt{v^2+2gh2} \\ vf=\sqrt{8.3^2+2(9.8)(3.35)} \\ vf=\sqrt{68.89+65.66} \\ vf=\sqrt{134.55} \\ vf\approx11.6m/s \end{gathered}[/tex]At the bottom of the slide the velocity of you and your friend together is 11.6 m/s
