How Does Adding Mass Affect Velocity in a Frictionless System?

[Trista]

Once again, I am pleading for help. Here is the problem and what I have done:
Gayle runs at a speed of 4.00 m/s and dives on a sled, initially at rest, on the top of a frictionless, snow-covered hill. After she has descended a vertical distance of 5.00m, her brother, who is initially at rest, hops on her back and they continue down the hill together. What is the final velocity if the total vertical distance is 15.0 m?
m1 = 50 kg
m2 = 5 kg
m3 = 30 kg
I figure I have to solve for the first part, where the girl jumps on the sled and goes 5 m:
KEi + PEi = KEf + PEf
| | | |
m1v1i +0 = m2v2f+mgh
50kg (4 m/s) = 55kg(v2f) + 55kg(9.8 m/s)(5 m)
then:
200 = 55kg(v2f) + 2695 then, subtracting 2695 from the right & left:
-2495 = 55kg (v2f) then, divide out the 55 from the right and then left:
-45.36 = v2f I suppose the negative sign shows they are traveling down the hill.
But then, assuming that my prior calculation is correct (which if it is, I want a ride on that sled), then I need to start over again except that my initial velocity would be v2f from the previous question, and I am looking for v3f, right? I hope this isn't too confusing, I'm trying to show that I've been working on this problem, and I would really like to understand it.
Thank you for your help!

 

[Andrew Mason]

Kinetic energy is not mv. It is $\int mvdv$

AM

 

[Trista]

ok, wait, this is the algebra section, right? that looks like an integral, and I'm doing algebra.

 

[daniel_i_l]

First of all kinetic energy is 1/2mv^2.
Also, after falling 5 meters the girl and sled have less potential energy than they did befor the fall, not more. You also should to use the conservation of momentum to find the new speed after the collision

 

[Andrew Mason]

ok, wait, this is the algebra section, right? that looks like an integral, and I'm doing algebra.

Ok. No calculus. You should know that KE = mv^2/2. Use your approach of conservation of energy.

There is a trick here though. The collision of the girl and the sled is not elastic. So you can use conservation energy only after the girl and sled starts moving together. As Daniel says, you have to use conservation of momentum to find that initial sled speed.

AM