A skier leaves the ramp of a ski jump with a velocity of 12.6m/s, 14.5° above the horizontal. The slope below is inclined downwards at 52.2°, and air resistance is negligible.
a)Calculate the distance from the ramp to where the jumper lands.
b) Calculate the velocity components just before the landing. Enter the x-component first and then the y-component.
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To solve this problem, we can use the kinematic equations to calculate the distance and velocity components of the skier.
a) To calculate the distance from the ramp to where the jumper lands, we need to find the horizontal range.
The horizontal velocity component can be calculated using the formula: vx = v * cos(θ), where v is the initial velocity and θ is the angle above the horizontal.
Given:
v = 12.6 m/s
θ = 14.5°
Substituting the values into the formula, we get:
vx = 12.6 m/s * cos(14.5°)
= 12.6 m/s * 0.9636
≈ 12.138 m/s
Now, let's calculate the time of flight. The time can be calculated using the formula: t = 2 * (v * sin(θ)) / g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the values into the formula, we get:
t = 2 * (12.6 m/s * sin(14.5°)) / 9.8 m/s^2
≈ 2.572 s
To find the horizontal range, we can multiply the horizontal velocity component by the time of flight:
Range = vx * t
= 12.138 m/s * 2.572 s
≈ 31.249 m
Therefore, the distance from the ramp to where the jumper lands is approximately 31.249 meters.
b) To calculate the velocity components just before the landing, we can use the formulas for projectile motion.
The final vertical velocity (vy) can be found using the formula: vy = v * sin(θ) - g * t
Substituting the values into the formula, we get:
vy = 12.6 m/s * sin(14.5°) - 9.8 m/s^2 * 2.572 s
≈ -15.488 m/s
Note that the negative value indicates the downward direction.
The final horizontal velocity (vx) remains constant throughout the motion and is equal to the initial horizontal velocity, which we calculated earlier:
vx = 12.138 m/s
Therefore, the velocity components just before the landing are approximately:
vx ≈ 12.138 m/s
vy ≈ -15.488 m/s