FBD of Skier

MAD of Skier

Using the principle of conservation of energy, the velocity of the skier at the end of the jump can be found. Energy is a simple way to find the velocity of any object if you know how far it falls. Of course, this method neglects air and surface friction which is lost energy. The principle of conservation was introduced in your basic physics course, and will studied in detail in the section on conservative forces and potential energy.

Equating the potential of the skier at the top of the jump with the kinetic energy at the bottom of the jump gives

     U_potential = U_kinetic

     hmg = 0.5mv²

If the velocity at the bottom of the jump is noted as v₂, then

The free-body and mass-acceleration diagrams are shown at the left. Summing the forces in the normal direction gives

     ΣF_n = ma_n

     N - mg = m v²/ρ

To solve for the normal force, N, the radius of curvature, ρ, must be determined at the end of the jump. From calculus, the radius of curvature for any continuous line function, y = f(x), is

     

For this problem,

Using the radius of curvature, the normal force exerted on the skier at the end of the jump is

     N = m v²/ρ + mg
        = 90 (17.15)²/20 + 90(9.8) = 2,210 N

To find the total acceleration of the skier, sum the forces in the tangential direction,

     ΣF_t = ma_t

     0 = ma_t

Thus, the acceleration at the end of the jump in the tangential direction is zero. The total acceleration is

     a = a_n e_n + a_t e_t
        = v²/ρ e_n + 0 e_t = 17.15²/20 e_n
       = 14.71 e_n m/s²