Unit 4 - Problem 5 ==> Ski Jump

Calculate the energy of the jumper leaving the ramp

• $$ E_{potential} = m*g*h = m(10)(90) $$
• $$ E_{lost-on-jump} = m*g*h = m(10)(10) $$
• $$ E_{leaving- jump} = E_{potential} - E_{lost-on-jump} = 900m - 100m = 800m$$

Calculate the jump velocities both horizontal and vertical

• $$ E_{leaving-jump} = E_{kinetic-energy} = 0.5 *m*V_{0-jump}^2 $$
• $$ 800m = 0.5*m*V_{0-jump}^2 $$
• $$ V_{0-jump}^2 = 1600 $$
• $$ V_{0-jump} =  40.0$$
• $$ \frac{V_{0-y-jump}}{V_0} = \sin\alpha = 0.5$$
• $$ V_{0-y-jump} = V_0 * .5 $$ 
• $$ V_{0-y-jump} = 20.0$$
• $$ \frac{V_{0-x-jump}}{V_0} = \cos\alpha = 0.866$$
• $$ V_{0-x-jump} = V_0 * .866 $$
• $$ V_{0-x-jump} = 34.64$$

Calculate: The energy on landing and the landing velocities

• $$ E_{landing} = E_{kinetic} + E_{jump-height} $$
• $$ E_{landing} = 800m + m(10)(10) = 900m $$
• $$ 0.5*m*V_{landing}^2 = 900m $$
• $$ V_{landing}^2 = 1800 $$
• $$ V_{landing} = 42.42640687119285 $$
• The horizontal jump velocity is a constant.
• $$ V_{landing}^2 = V_{0-x-jump}^2 + V_{y-landing}^2$$
• $$ V_{landing}^2 - V_{0-x-jump}^2 = V_{y-landing}^2$$
• $$ V_{y-landing}^2 = 1800 - 1199.9296 = 1800 - 1200 = 600 $$
• $$ V_{y-landing}  = 24.49489742783178 $$

Calculate: Time to peak using vertical component of jump velocity

• $$ t_{jump-to-peak} = \frac{V_{peak} - V_{0-y-jump}}{a} $$
• $$ t_{jump-to-peak} = \frac{-20}{g} $$
• $$ t_{jump-to-peak} = 2.0$$

Calculate: Time from peak using the vertical component of the landing velocity

• $$ t_{peak-to-landing} = \frac{V_{peak} - V_{y-landing}}{a} $$
• $$ t_{peak-to-landing} = \frac{- V_{y-landing}}{a}$$
• $$ t_{peak-to-landing} = \frac{24.49489742783178}{g} = 2.449489742783178 $$

Calculate:Total time is sum of time up to peak plus time down to ground

• $$ t_{total} = t_{jump-to-peak} + t_{peak-to-landing}$$
• $$ t_{total} = 2.0 + 2.449489742783178 = 4.449489742783178 $$

Calculate d based on total time and horizontal velocity at jump

• $$ d_{total} = V_{0-x-jump} * t_{total} $$
• $$ d_{total} = 34.64 * 4.449489742783178 $$
• $$ d_{total} = 154.13032469000927 = 154 $$