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 = mgcos Θ - n

ΣFy = ma = mgcos

So...

ΣFy = ma = mgcos

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

**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 = ½v² + gh

**<---**Let's plug in the values we got earlier for v²and h

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 ☺)