Law of Conservation of Energy of a downhill skier

[darkmasterz8]

1. A skier is pushed from the top of a hill so that he starts moving down the hillside sloped at 27.6° to the horizontal with an initial speed of 0.434 m/s. After traveling 80.4 m, he reaches the bottom of the valley. Due to inertia, he then continues 70.4 m up another hillside sloped at 20.7° to the horizontal. What is the skier's speed when he reaches the top of the hill? Assume that you can neglect friction.
2. KE(b)+PE(b)=KE(a)+PE(a)
KE(b)+PE(b)=KE(a)+PE
1/2*m*(.434^2)+m*9.8*80.4=1/2*m*v^2+m*9.8*70.4---X out all m since they're the same both sides

.09+787.9=1/2*v^2+689.9
788=1/2*v^2+689.9
98.11=1/2*v^2
196=v^2
14=v

I already know the answer is 15.6 but I don't seem to get it through the work I've done. What I'm confused on is if 80.4 and 70.4 are the height? The problem makes it sound more like distance which doesn't make sense to me. I'm pretty sure that is the formula for law of conservation of energy.

 

[hy23]

yes 80.4 and 70.4 are the distances that the person travels...not the height, you will have to use the angle of inclination to find the actual height

and yes, you will encounter a lot of problems that give you distances to make your life harder

 

[darkmasterz8]

Well to use angle of inclination to find the height for initial, you do cosine 27.6=x/80.4 where x = 71.2 meters. The same thing with the after part. But when I plug in the height into the above equation, I still don't get it.

I'm not sure which of the three to use; cosine/tangent/sine because I don't know where the angle belongs. It says 27.6 to the horizontal so I assume the angle is at the top and you use cosine to find the adjacent given the hypotnuse of 80.4.

EDIT:
OOO I think I just got it. It's sine to get it.

Sine 27.6=x/80.4 where x=37.25
Sine 20.7=x/70.4 where x=24.77

Sub it in the above equation and I get 15.6 m/s.

Thanks for telling me about height/distance :)

 

[hy23]

draw a triangle with a right angle, the distance is the hypotenuse and the angle of inclination is opposite the height

this might be a good way to remember it ;)

 

[rwisz]

I would like to clarify a few things about the problem before providing a response. The given problem does not specify the mass of the skier, which is necessary to solve for the final speed. Also, the given values of 80.4 m and 70.4 m are most likely the horizontal distances traveled by the skier, not the heights. This is because the problem states that the skier is moving downhill and then uphill, suggesting a change in height. If these were heights, the skier would end up at the same elevation as the starting point.

Assuming the mass of the skier is known and the given values are horizontal distances, the law of conservation of energy can be applied to solve for the final speed. The equation you have written is correct, but it is missing the mass term. It should be:

KE(b) + PE(b) = KE(a) + PE(a) + Wfr

Where Wfr is the work done by friction, which is assumed to be negligible in this problem.

Using this equation and substituting the given values, we get:

1/2 * m * (0.434 m/s)^2 + m * 9.8 m/s^2 * 80.4 m = 1/2 * m * v^2 + m * 9.8 m/s^2 * 70.4 m

Simplifying and solving for v, we get:

v = √(0.434^2 + 2 * 9.8 * (80.4 - 70.4)) = 15.6 m/s

Therefore, the skier's speed when he reaches the top of the hill is 15.6 m/s. This is slightly different from the given answer of 15.6 m/s, which could be due to rounding off errors.