Question

A skier of mass 75 kg starts from rest and skis down a curved slope. The drop in altitude from the top to the bottom of the slope is 266 m, and the skier travels 86 m horizontally. Neglecting air resistance and friction from the snow, find the skier's speed (in m/s ) at the bottom of the slope.

Answer 1

To find the skier's speed at the bottom of the slope, we can use the principle of conservation of energy. The skier's speed at the bottom of the slope is approximately 37.19 m/s.

Step-by-step explanation:

To find the skier's speed at the bottom of the slope, we can use the principle of conservation of energy. At the top of the slope, the skier has gravitational potential energy, which is converted to kinetic energy at the bottom of the slope. Since there is no air resistance or friction, the energy is conserved. The gravitational potential energy at the top can be calculated using the formula MGH, where m is the mass of the skier, g is the acceleration due to gravity, and h is the drop in altitude. The kinetic energy at the bottom can be calculated using the formula 1/2mv^2, where v is the speed of the skier at the bottom.

Let's plug in the given values: m = 75 kg, g = 9.8 m/s^2, h = 266 m. The gravitational potential energy at the top is mgh = 75 kg * 9.8 m/s^2 * 266 m = 194,670 J. Since the energy is conserved, this equals the kinetic energy at the bottom of the slope: 1/2mv^2 = 194,670 J. Solving for v, we get v = √(2 * 194,670 J / 75 kg) ≈ 37.19 m/s.

Answer 2

Using the conservation of energy principle and the formula for potential energy (PE = mgh), the skier of mass 75 kg traveling from an altitude drop of 266 m will have a final speed of about 72.3 m/s at the bottom of the slope, having neglected friction and air resistance.

Step-by-step explanation:

To determine the skier's speed at the bottom of the slope, we can use the conservation of energy principle. The skier's potential energy at the top will be converted into kinetic energy at the bottom, since we are neglecting air resistance and friction.

The potential energy (PE) at the top of the slope can be calculated using the formula PE = mgh, where m is the mass of the skier (75 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the drop in altitude (266 m). So, PE = 75 kg × 9.8 m/s² × 266 m.

The kinetic energy (KE) at the bottom of the slope is given by KE = 0.5 × m × v², where v is the speed of the skier we need to find. At the bottom of the slope, the potential energy converted to kinetic energy, so PE = KE. By setting the equations equal to each other and solving for v, we can find the skier's speed at the bottom.

After calculations, the skier's speed at the bottom of the slope is found to be the square root of (2 × g × h). Plugging in the values, the speed v = √(2 × 9.8 m/s² × 266 m), which gives a final velocity of approximately 72.3 m/s.