A ski jumper starts with a horizontal take-off velocity of 25m/s and lands on a
straight landing hill inclined at 30 0 . Determine (a) the time between take-off and
landin. (b) The length 'd' of the jump. (c) The maximum vertical distance between
the jumper and the landing hill.
(a) The time between take-off and landing can be calculated using the equation:
t = v/g
where v is the initial velocity and g is the acceleration due to gravity (9.8 m/s2).
t = 25/9.8 = 2.55 s
(b) The length of the jump can be calculated using the equation:
d = v2sin2θ/2g
where v is the initial velocity, θ is the angle of the landing hill (30°), and g is the acceleration due to gravity (9.8 m/s2).
d = (252sin2(30°))/2(9.8) = 24.7 m
(c) The maximum vertical distance between the jumper and the landing hill can be calculated using the equation:
h = v2sin2θ/2g
where v is the initial velocity, θ is the angle of the landing hill (30°), and g is the acceleration due to gravity (9.8 m/s2).
h = (252sin2(30°))/2(9.8) = 12.3 m
To solve these problems, we can use the equations of motion for projectile motion. Let's go step by step.
(a) The time between take-off and landing can be determined using the equation:
Δy = (Voy * t) + (0.5 * g * t^2),
where Δy is the vertical displacement, Voy is the initial vertical velocity, t is the time, and g is the acceleration due to gravity.
Since the ski jumper starts with only a horizontal take-off velocity, Voy = 0. So the equation becomes:
Δy = 0.5 * g * t^2.
The vertical displacement Δy can be found using trigonometry. The opposite side of the triangle formed by the landing hill is Δy, the hypotenuse is the distance traveled (d), and the angle is 30 degrees.
Δy = d * sin(30).
Combining these equations, we get:
d * sin(30) = 0.5 * g * t^2.
Now solving for time (t):
t = sqrt((2 * d * sin(30)) / g).
(b) To find the length of the jump (d), we can use the equation:
d = Vx * t,
where Vx is the horizontal velocity and t is the time. Since the ski jumper has a horizontal take-off velocity of 25 m/s, we can substitute it into the equation:
d = 25 * t.
Substituting the value of t from part (a), we get:
d = 25 * sqrt((2 * d * sin(30)) / g).
Simplifying this equation will give us the length of the jump (d).
(c) The maximum vertical distance between the jumper and the landing hill is equal to the vertical displacement (Δy). In this case, Δy can be found using the equation:
Δy = 0.5 * g * t^2.
Substituting the value of t from part (a), we can find the maximum vertical distance.
These calculations involve some mathematical manipulation, so let's solve them step by step.
Step 1: Solve for time (t) using the equation from part (a):
Δy = d * sin(30) = 0.5 * g * t^2.
t = sqrt((2 * d * sin(30)) / g).
Step 2: Solve for the length of the jump (d) using the equation from part (b):
d = 25 * t = 25 * sqrt((2 * d * sin(30)) / g).
Step 3: Solve for the maximum vertical distance (Δy) using the equation from part (c):
Δy = 0.5 * g * t^2.
These equations are interdependent, so we need to solve them iteratively.
To solve this problem, we need to break it down into different parts.
(a) Finding the time between take-off and landing:
The horizontal motion of the ski jumper is unaffected by the inclined landing hill. Therefore, the horizontal component of the velocity remains constant at 25 m/s throughout the jump. We can use this information to calculate the time taken by the jumper.
The horizontal distance traveled by the jumper can be calculated using the formula:
distance = velocity * time
In this case, the distance traveled horizontally is the length of the jump, which we'll find later. The initial velocity is 25 m/s. Therefore, the equation becomes:
distance = 25 * time
We need to solve for time, so rearranging the equation:
time = distance / 25
(b) Finding the length 'd' of the jump:
The length of the jump can be calculated using the horizontal component of the velocity and the time calculated in step (a).
length = velocity * time
Given that the horizontal velocity is 25 m/s, and we calculated the time in step (a), substitute these values into the equation:
length = 25 * time
(c) Finding the maximum vertical distance between the jumper and the landing hill:
To find the maximum vertical distance, we need to consider the vertical motion of the ski jumper.
At take-off, the ski jumper has an initial vertical velocity of 0 m/s, and the only force acting on the jumper is gravity. The jumper will rise and then fall back down due to gravity.
The time it takes the jumper to reach the maximum height can be found using the formula:
time_to_max_height = vertical_velocity / gravitational_acceleration
In this case, the vertical velocity is 0 m/s, and the gravitational acceleration is 9.8 m/s^2. Therefore:
time_to_max_height = 0 / 9.8 = 0 seconds
Since the time to reach the maximum height is 0 seconds, it means the jumper reaches it instantly. The maximum height is the vertical distance covered in this time.
The maximum vertical distance can be found using the formula for vertical displacement:
vertical_distance = (initial_vertical_velocity * time) + (0.5 * gravitational_acceleration * time^2)
The initial vertical velocity is 0 m/s, and time is 0 seconds. Therefore:
vertical_distance = 0 + 0 = 0 meters
Hence, the maximum vertical distance between the jumper and the landing hill is 0 meters.
To summarize:
(a) The time between take-off and landing is distance / 25.
(b) The length 'd' of the jump is 25 * time.
(c) The maximum vertical distance between the jumper and the landing hill is 0 meters.