Skiing both ways conservation of energy/1D kinematic

[J827]

skiing both ways...conservation of energy/1D kinematic

Homework Statement

A skier starts from rest at the top of a 45.0-m-high hill, skis down a 30° incline into a valley, and continues up a 40.0-m-high hill. The heights of both hills are measured from the valley floor. Assume that friction is negligible and ignore the effect of the ski poles.

a. How fast is the skier moving at the bottom of the valley?

b. What is the skier’s speed at the top of the second hill?

Homework Equations

Equation A
KE_i + PE_i = KE_f + PE_f

or

Equation B
v_f² = v_i² + 2ad

The Attempt at a Solution

I know that the angle is irrelevant because only gravity is affecting his movement.

I understand part a. Using Equation A, initial KE and final PE are zero. Final velocity is 30m/s. I get the same answer using Equation B. I confirmed this answer with the solution manual.

I am getting stuck on part b. Here is what I have tried so far:

Option 1a:
starting from the valley, using Equation A
Initial PE = 0, because you are in the valley.
Mass factors out of equation because is in three remaining terms.

(0.5 * 30²) = (0.5 * V_f²) + (9.8 * 40)
v_f = 10.8 m/s

Option 1b:
same as 1a, but using -9.8 since he's going in a different direction
v_f = 41 m/s

Option 2:
starting from the valley, using Equation B
v_f² = (30²) + (2 * -9.8 * 40)
vf = 10.8 m/s

Option 3a (as taken from official solution manual):
starting from the top of the first hill, using Equation A
initial KE = 0
mass factors out of equation

gh_i = 0.5v_f² + gh_f
(9.8*45) = (0.5v_f²) + (9.8 * 40)
v_f = 9.9 m/s

Option 3b
(because I don't understand why the same sign for g would be used if he is going in 2 different directions)
(9.8 * 45) = (0.5v_f²) + (-9.8 * 40)
v_f = 41 m/s

 

[rude man]

Your problem is roundoff error. You did not calculate your v at the bottom of the hill correctly. It's less than 30 m/s.

 

[J827]

*headdesk*

thanks.

 

[CWatters]

Edit: looks like I took too long thinking about why g is positive. Anyway..

I make the velocity at the bottom 29.7m/s not 30m/s. Makes all the difference.

Otherwise ..

1a and 2 and 3a are all valid approaches and all give the same answer 9.9m/s

1b and 3b are wrong.

I've not got a good explanation for why g is positive accept to say that as he climbs the other hill he clearly must gain PE.

 

[rude man]

g does not change - ever!

 

[haruspex]

(because I don't understand why the same sign for g would be used if he is going in 2 different directions)

It all depends on how you define the positive direction. If up is positive in both parts then g is negative in both and the vertical distance is negative in a), positive in b).

 

[rude man]

If I drop something it falls toward Earth. It doesn't matter if I'm going uphill or downhill.

 

[J827]

while the magnitude of g never changes, how it affects your velocity will depending on how you have defined direction. if I am skiing downhill, my velocity will increase at a rate of 9.8 m/s². As I ski uphill, however, my velocity will decrease at a rate of 9.8 m/s².

I've not got a good explanation for why g is positive accept to say that as he climbs the other hill he clearly must gain PE.

that makes sense & explains why using when Equation A 9.8 is positive, while using Equation B 9.8 is negative.

 

[haruspex]

If I drop something it falls toward Earth. It doesn't matter if I'm going uphill or downhill.

Was that in response to my post? I don't see how what I said contradicts that. It's g either way, but depending on how you define the positive direction, g is either +9.8ms^-2 or -9.8ms^-2.

 

[rude man]

Was that in response to my post? I don't see how what I said contradicts that. It's g either way, but depending on how you define the positive direction, g is either +9.8ms^-2 or -9.8ms^-2.

I know you well enough to know that you know that the sign of g does not change once the coordinate system is defined!

No, I sensed the OP thought g was one sign going down & the other going up.

 

[J827]

No, I sensed the OP thought g was one sign going down & the other going up.

nope, I'm good. :)