Sunday, July 5, 2015

Skier on a Snowball (Physics Problem Solved)

Written by Rebekah

Let’s solve the following physics problem:

“A skier starts at the top of a very large, frictionless snowball, with a very small initial speed, and skis straight down the side. At what point does he lose contact with the snowball and fly off at a tangent? That is, at the instant he loses contact with the snowball, what angle does a radial line from the center of the snowball to the skier make with the vertical?”

Let’s begin by making a more detailed picture.

As you can all see, our drawing is pretty involved, so let’s talk a little about it.  First notice that we are looking for the angle Θ.  Because we basically have two parallel lines cut by a transversal, we can use Θ in our body diagram.  Then we can break the force acting on the skier into its components (notice the coordinate I chose for this).

So then, this means that the force acting in the y direction is: 

ΣFy = ma = mgcosΘ - n

Loosing contact implies that the normal force, n, equals  0


ΣFy = ma = mgcosΘ

But remember that the formula for centripetal acceleration is v²/R (R being the radius) . 

Rewriting the equation for F = ma, we get 

mgcosΘ = mv²/R

We can divide both sides by m, and then solve for  

v² = RgcosΘ <-----This will come in handy later.

From this drawing we can easily see that cosΘ = h/R

Solving for h, we see that: h = RcosΘ<---This will come in handy later.

But how does all this come together? Through conservation of energy!

Conservation of energy implies:

0 + mgR = ½mv² + mgh
        gR = ½ + gh<---Let's plug in the values we got earlier for v²and h
        gR½RgcosΘ + gRcosΘ
          1 = ½cosΘ + cosΘ
          1 = 3/2 cosΘ
      2/3 =  cosΘ
arccos(2/3) = 48.2°

So there you have it, the answer is Θ = 48.2° (isn't that nice ☺)