#### Problem

Use Newtonian mechanics to derive the equation of motion

(1)for the “snowboarder” described in class.

#### Solution

To derive the equation using Newtonian mechanics, we first draw a free body diagram of the snowboarder of mass $m$ riding along the curve $f(x)$ under the influence of gravity $g$:

As seen in the slick Microsoft Paint free body diagram above, the gravitational force $mg$ can be split into components normal and tangent to the curve $f(x)$ at any given point. The normal component of gravity will be exactly canceled by the normal force $N$ exerted by the curve (hill) on the mass (snowboarder). That leaves the tangential force (call it $F_t$) to cause the mass to move along the curve according to Newton's second law,

(2)Before we go any further, note that the angle $\theta$ can be derived as follows:

(3)Also, note that the velocity $v$ of the mass can be calculated as follows:

(5)Returning to the free body diagram, we can come up with an expression for the tangential force to use on the left side of Newton's second law.

(6)The sign of the tangential force is negative because it acts “against” the slope of the hill.

We now plug everything in to Newton's second law and simplify:

(7)And finally:

(11)Scott agrees!