A skier leaves the ramp of a ski jump with a velocity of
v = 8.0 m/s
at
𝜃 = 28.0°
above the horizontal, as shown in the figure. The slope where she will land is inclined downward at
𝜑 = 50.0°,
and air resistance is negligible.
To determine the landing distance of the skier, we can break down the given information and use the principles of projectile motion. Here's how we can calculate it:
1. Split the initial velocity into its horizontal and vertical components:
- The horizontal component (v_x) remains constant throughout the motion and can be found using the formula: v_x = v * cos(𝜃)
- The vertical component (v_y) will change due to the effects of gravity and can be found using the formula: v_y = v * sin(𝜃)
2. Determine the time of flight (t) for the skier. Since we are neglecting air resistance, the time it takes to reach the peak of the jump will be the same as the time it takes to descend to the landing slope. You can use the formula: t = 2 * v_y / g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
3. Calculate the horizontal distance traveled (d) by the skier using the formula: d = v_x * t
4. Finally, calculate the vertical distance (h) from the horizontal plane of the launch to the landing slope using the formula: h = d * tan(𝜑)
5. The landing distance is given by the formula: landing distance = d + h
By plugging in the given values into these equations, you can find the landing distance of the skier.