When the skier leaves the end of the ski jump, they have an initial horizontal velocity of 25.0 m/s. This means that their horizontal velocity remains constant throughout the motion. The vertical distance fallen by the skier is 3.10 m.
Let's analyze the skier's motion separately in the horizontal and vertical directions:
Horizontal Motion: Since there are no horizontal forces acting on the skier (assuming negligible air resistance), their horizontal velocity of 25.0 m/s remains constant. Therefore, the skier will continue moving horizontally at a constant speed until they land.
Vertical Motion: The vertical motion of the skier is affected by the force of gravity. When the skier leaves the ski jump, they have an initial vertical velocity of 0 m/s. The only force acting on the skier in the vertical direction is gravity, which causes the skier to accelerate downwards at a rate of approximately 9.8 m/s² (assuming no air resistance).
Using the kinematic equation for vertical motion:
Δy = v₀y * t + (1/2) * a * t²
where Δy = vertical displacement (3.10 m) v₀y = initial vertical velocity (0 m/s) a = acceleration due to gravity (-9.8 m/s²) t = time
Rearranging the equation:
Δy = (1/2) * a * t²
Solving for t:
t = sqrt((2 * Δy) / a)
Plugging in the values:
t = sqrt((2 * 3.10 m) / (-9.8 m/s²)) t ≈ 0.79 s
Therefore, the skier will take approximately 0.79 seconds to fall through the vertical distance of 3.10 meters.
In conclusion, the skier will continue moving horizontally at a constant speed of 25.0 m/s while falling vertically through a distance of 3.10 meters. They will eventually land after 0.79 seconds of falling time.