A ski jumper acquires a speed of 108.6 km/hr by racing down a steep hill. He then lifts off into the air from a horizontal ramp. Beyond this ramp, the ground slopes downward at an angle of 45 degrees. Assuming the skier is in free-fall motion after he leaves the ramp, at what distance d down the slope does the skier land?
To determine the distance the skier lands, we need to find the horizontal distance traveled by the skier while in free-fall motion.
First, we need to convert the initial speed of the skier from kilometers per hour to meters per second. We can do this by dividing the speed by 3.6:
108.6 km/hr / 3.6 = 30.17 m/s (rounded to two decimal places)
Next, we can break down the initial speed into its horizontal and vertical components. The vertical component of the speed will determine how long the skier will be in the air.
The vertical component of the speed can be found using trigonometry:
Vertical component = speed * sin(angle)
Vertical component = 30.17 m/s * sin(45 degrees)
Vertical component = 30.17 m/s * 0.7071
Vertical component = 21.35 m/s (rounded to two decimal places)
The horizontal component of the speed will determine the distance the skier will travel:
Horizontal component = speed * cos(angle)
Horizontal component = 30.17 m/s * cos(45 degrees)
Horizontal component = 30.17 m/s * 0.7071
Horizontal component = 21.35 m/s (rounded to two decimal places)
Since the skier is in free-fall motion, the time of flight can be determined using the vertical component of speed and the acceleration due to gravity (9.8 m/s^2):
Time of flight = (2 * vertical component) / acceleration due to gravity
Time of flight = (2 * 21.35 m/s) / 9.8 m/s^2
Time of flight = 4.35 seconds (rounded to two decimal places)
We can then calculate the horizontal distance traveled by the skier using the horizontal component of speed and the time of flight:
Distance = horizontal component * time of flight
Distance = 21.35 m/s * 4.35 s
Distance = 92.77 meters (rounded to two decimal places)
Therefore, the skier lands at a horizontal distance of approximately 92.77 meters down the slope.
To find the distance d down the slope where the skier lands, we can break down the problem into vertical and horizontal components. Firstly, let's convert the speed of the skier from km/hr to m/s.
Given:
Speed of the skier (v) = 108.6 km/hr
To convert km/hr to m/s, we use the conversion factor 1 km/hr = 0.2778 m/s.
So, v = 108.6 km/hr * (0.2778 m/s / 1 km/hr) = 30.16 m/s.
Since the skier is in free-fall motion, the horizontal component of his velocity remains constant at 30.16 m/s throughout the entire motion.
Now, let's consider the vertical component of his motion. The skier takes off from a horizontal ramp and the ground slopes downward at an angle of 45 degrees. So, the vertical component of his velocity is affected by gravity.
The vertical component of the initial velocity (v₀y) can be found using the formula:
v₀y = v * sin(θ)
Where:
v = initial velocity (30.16 m/s)
θ = angle of the slope (45 degrees)
Substituting in the values, we get:
v₀y = 30.16 m/s * sin(45°) = 21.34 m/s
Now, we can use this vertical component of velocity to find the time (t) it takes for the skier to land. In free-fall motion, the time of flight can be found using the formula:
t = 2 * v₀y / g
Where:
v₀y = initial vertical velocity (21.34 m/s)
g = acceleration due to gravity (9.8 m/s²)
Substituting in the values, we get:
t = 2 * 21.34 m/s / 9.8 m/s² = 4.34 s
Now, we need to find the horizontal distance (dx) covered by the skier during this time.
The horizontal distance can be found using the formula:
dx = v * t
Where:
v = horizontal velocity (30.16 m/s)
t = time (4.34 s)
Substituting in the values, we get:
dx = 30.16 m/s * 4.34 s = 130.98 m
Therefore, the skier will land at a distance d down the slope of approximately 130.98 meters.
To find the distance at which the skier will land down the slope, we need to analyze the horizontal and vertical components of the skier's motion separately.
Let's start by breaking down the initial velocity of the skier into horizontal and vertical components.
Given:
- Initial speed (along the slope) = 108.6 km/hr
- Angle of the downward slope = 45 degrees
Step 1: Convert the initial speed from km/hr to m/s.
To convert from km/hr to m/s, divide by 3.6
Initial speed (along the slope) = 108.6 km/hr ÷ 3.6 = 30.17 m/s
Step 2: Calculate the horizontal and vertical components of the velocity.
The horizontal component of the velocity remains constant throughout the motion, while the vertical component changes due to gravity.
Horizontal component (V_h) = Initial speed (along the slope) * cos(angle of the slope)
= 30.17 m/s * cos(45 degrees)
≈ 21.37 m/s
Vertical component (V_v) = Initial speed (along the slope) * sin(angle of the slope)
= 30.17 m/s * sin(45 degrees)
≈ 21.37 m/s
Now that we have the initial vertical velocity (V_v) and we know the skier is in free fall, we can use the equations of motion to find the distance (d) the skier will land down the slope.
Step 3: Determine the time of flight (t) of the skier.
In free fall, the time of flight can be found using the equation:
V_v = g * t,
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Rearranging the equation to solve for t:
t = V_v / g
= 21.37 m/s / 9.8 m/s^2
≈ 2.18 s
Step 4: Calculate the horizontal distance traveled (d) by the skier using the time of flight (t).
The horizontal distance traveled can be found using the equation:
d = V_h * t
Substituting the values:
d = 21.37 m/s * 2.18 s
≈ 46.55 m
Therefore, the skier will land approximately at a distance of 46.55 meters down the slope.