To solve this problem, we can use the principles of projectile motion.
First, we can break down the initial velocity of the ski jumper into its horizontal and vertical components. Since the ski jumper leaves the ski track moving horizontally with a speed of 21 m/s, the initial horizontal velocity (Vx) is 21 m/s. The initial vertical velocity (Vy) is 0 m/s because there is no vertical component of the initial velocity.
Next, we need to find the time it takes for the ski jumper to land. We can use the equation of motion in the vertical direction:
h = (1/2)gt^2
where h is the vertical distance traveled and g is the acceleration due to gravity. Since the ski jumper is landing on an incline, we can consider the height h as the vertical distance from the point where the ski jumper leaves the ski track to the point where she lands on the incline.
The vertical distance, h, can be calculated using the trigonometric relationship:
h = x * tan(θ)
where x is the horizontal distance traveled along the incline and θ is the angle of the incline. Given that θ = 49°, we can solve for h:
h = x * tan(49°)
Now, we can substitute this expression for h into the equation of motion:
x * tan(49°) = (1/2)gt^2
Since the time t it takes for the ski jumper to land is the same for the horizontal and vertical motion, we can solve for t:
x * tan(49°) = (1/2)g * t^2
t^2 = (2 * x * tan(49°)) / g
Taking the square root of both sides, we get:
t = sqrt((2 * x * tan(49°)) / g)
Next, we can use the equation of motion in the horizontal direction:
x = Vx * t
Substituting the expression for t, we get:
x = Vx * sqrt((2 * x * tan(49°)) / g)
Now we can solve for x:
x^2 = (Vx^2 * 2 * x * tan(49°)) / g
x^2 * g = Vx^2 * 2 * x * tan(49°)
x * g = Vx^2 * 2 * tan(49°)
Simplifying further, we get:
x = (Vx^2 * 2 * tan(49°)) / g
Now we can substitute the given values into the equation to find the distance x:
x = (21^2 * 2 * tan(49°)) / 9.8
x ≈ 115.97 m
Therefore, the ski jumper travels approximately 115.97 m along the incline before landing.
A ski jumper travels down a slope and leaves
the ski track moving in the horizontal direction with a speed of 21 m/s as in the figure.
The landing incline below her falls off with a
slope of θ = 49◦
.
The acceleration of gravity is 9.8 m/s
2
Calculate the distance d she travels along
the incline before landing.
Answer in units of m.
1 answer