Ski Jumper Speed and Distance


A ski jumper starts from rest at the top of a ski jump which has a vertical drop to the take off point of 46 m. The take off point is 4 m above the top of a landing slope which is 30 degrees below horizontal. Assuming that we can neglect friction and air resistance, a real stretch here, determine his speed at the take off point and assuming he leaves the take off point horizontally, determine the distance down the landing slope from the foot of the ski jump under the take off point to the point of the skier's landing. The skier has a mass of 70 kg.


The skier's energy is all potential at the top of the jump. When he gets to the take off point, 46 meters worth of gravitational potential energy will have been converted to kinetic energy. Potential energy is m*g*h, in this case 70 kg * 9.8 m/s2 * 46 m = 31556 Joules. We know that 1/2*70*v2=31556 so v2=902 m2/s2, so v=30 m/s.

With a horizontal launch at 30 m/s, the horizontal position of the skier as a function of time is given by Dh=30*t. The vertical position is given by Dv=1/2*9.8*t2. The vertical distance to landing as a function of time is 4+30*t*sin(30) = 19*t. When the vertical distance to landing is equal to the vertical distance fallen, the skier lands. When 4.9*t2=19*t, the jump is over. This occurs at time t=19/4.9 = 3.88 seconds. In that time the skier has gone 30*3.88 meters downrange, 116.4 meters. The distance to the point of impact measured along the landing slope will be 116.4/cos(30) = 134 meters.

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