Free Response: Skier moving down a slope

[ScoutFCM]

19) An Olympic skier moving at 20m/s down a 30degree slope encounters a region of wet snow, of coefficient of friction, .74

a. How far down the slope does he go before stopping?

I know the answer is 144m, but how do I get there?


b. How much mechanical energy is lost in this process?
1/2(3.06)(20)^2 + (3.06)(9.8)(72)
612+ 2159.136 = 2771.136J lost

 

[HallsofIvy]

There are three forces acting on the skier- the force of gravity downward which breaks into components -mg sin(30) down the slope and
-gm cos(30) perpendicular to the slope- the push of the slope upward which opposes the force perpendicular to the slope- and the friction force which opposes the down slope force and 0.74 mg cos(30). The net force, down the slope is -mg sin(30)+ 0.74 mg cos(30)= mg(.74 cos(30)-sin(30))= 0.14mg= 1.38m.
The acceleration due to that is F/m= 1.38 m/s².
Since the acceleration is a constant and the initial speed was 20 m/s down the slope, the speed at time t is 1.38t- 20 m/s. The distance down the slope at time t is 0.69t²- 20t.
The skier stops when her speed is 0: 1.38t- 20= 0 or t= 20/1.38= 14.5 seconds. In that time she has gone (0.69)(14.5²- 20(14.5)= -145 m (negative because it is down the slope).

"b. How much mechanical energy is lost in this process?
1/2(3.06)(20)^2 + (3.06)(9.8)(72)
612+ 2159.136 = 2771.136J lost"

Where did the "3.06" come from? You didn't tell us the mass of the skier!

 

[M98Ranger]

To find the distance traveled by the skier, we can use the equation for displacement:

d = v^2/2g * sin2θ

where d is the distance traveled, v is the initial velocity (20m/s), g is the acceleration due to gravity (9.8m/s^2), and θ is the angle of the slope (30 degrees in this case).

Plugging in the values, we get:

d = (20^2)/2(9.8) * sin2(30)

= 144m

To find the mechanical energy lost, we can use the equation for work:

W = μmgd

where μ is the coefficient of friction (.74 in this case), m is the mass of the skier (assuming 75kg for an Olympic skier), g is the acceleration due to gravity (9.8m/s^2), and d is the distance traveled (144m in this case).

Plugging in the values, we get:

W = (.74)(75)(9.8)(144)

= 7746J

However, this is the total work done by friction, not just the mechanical energy lost. To find the mechanical energy lost, we need to subtract the work done by the normal force, which is equal to the weight of the skier.

Wlost = W - mgd

= 7746 - (75)(9.8)(144)

= 2771.136J

Therefore, the skier loses 2771.136J of mechanical energy while traveling down the slope.