A skateboarder in a death defying stunt decides to launch herself from a ramp on a hill. The skateboarder leaves the ramp at a height of 1.4m above the slope, traveling 15m/s and at an angle of 40 degrees to the horizontal. The slope is inclined at 45 degrees to the horizontal. a). How far down the slope does the skateboarder land? b). How long is the skateboarder in the air? c). With what velocity does the skateboarder land on the slope?
To calculate the different aspects of the skateboarder's jump, we can break it down into different steps. Let's go through each step one by one:
Step 1: Find the horizontal and vertical components of the initial velocity.
The skateboarder has an initial velocity of 15 m/s at an angle of 40 degrees to the horizontal. We can find the horizontal and vertical components, denoted as Vx (horizontal) and Vy (vertical).
Vy = V * sinθ
Vy = 15 m/s * sin(40°)
Vy = 15 m/s * (0.6428)
Vy ≈ 9.64 m/s
Vx = V * cosθ
Vx = 15 m/s * cos(40°)
Vx = 15 m/s * (0.766)
Vx ≈ 11.49 m/s
Step 2: Calculate the time of flight.
To find the time the skateboarder is in the air, we can use the vertical component of velocity (Vy) and the acceleration due to gravity (g = 9.8 m/s²).
At the highest point (when the skateboarder is momentarily weightless), Vy = 0.
0 = Vy - g * t
t = Vy / g
t = 9.64 m/s / 9.8 m/s²
t ≈ 0.9847 s
Since the skateboarder is in the air for half of the total time of flight, the time of flight (t_flight) is:
t_flight = 2 * t
t_flight ≈ 2 * 0.9847 s
t_flight ≈ 1.9694 s
Step 3: Calculate the horizontal distance traveled.
The horizontal distance traveled (d) can be found using the horizontal component of velocity (Vx) and the time of flight (t_flight).
d = Vx * t_flight
d = 11.49 m/s * 1.9694 s
d ≈ 22.65 m
Therefore, the skateboarder lands approximately 22.65 meters down the slope.
Step 4: Calculate the velocity at landing.
To find the velocity at landing (V), we can use the time of flight (t_flight) and the vertical component of velocity (Vy) at landing.
V = Vy + g * t_land
Since Vy = 0 at the highest point, Vy at landing is the vertical component of velocity after time t_flight.
Vy_land = -g * t_flight
Vy_land = -9.8 m/s² * 1.9694 s
Vy_land ≈ -19.20 m/s
To get the magnitude of the velocity at landing, we use the Pythagorean theorem:
V_land = √(Vx² + Vy_land²)
V_land = √((11.49 m/s)² + (-19.20 m/s)²)
V_land ≈ √(131.8901 + 369.792) ≈ √501.6821
V_land ≈ 22.40 m/s
Therefore, the skateboarder lands with a velocity of approximately 22.40 m/s on the slope.
Summary:
a) The skateboarder lands approximately 22.65 meters down the slope.
b) The skateboarder is in the air for approximately 1.9694 seconds.
c) The skateboarder lands with a velocity of approximately 22.40 m/s on the slope.
To answer these questions, we need to break down the motion of the skateboarder into horizontal and vertical components. We can use the kinematic equations of motion to find the answers.
a) First, let's calculate how far down the slope the skateboarder lands.
We'll need to find the horizontal component of the skateboarder's initial velocity. Given that the skateboarder is traveling at 15 m/s at an angle of 40 degrees to the horizontal, we can calculate the horizontal component using the equation:
Horizontal component = initial velocity * cos(angle)
Horizontal component = 15 m/s * cos(40 degrees)
Now, let's calculate the time it takes for the skateboarder to land. To do this, we need to find the time it takes for the skateboarder to reach the maximum height and then double that time.
The vertical component of the skateboarder's initial velocity can be found using the equation:
Vertical component = initial velocity * sin(angle)
Vertical component = 15 m/s * sin(40 degrees)
Now, we can find the time it takes to reach the maximum height using this vertical component and the acceleration due to gravity (9.8 m/s^2). The formula to find time is:
time = (vertical component) / acceleration due to gravity
time = (15 m/s * sin(40 degrees)) / 9.8 m/s^2
Once we have the time to reach the maximum height, we can double it to find the total time in the air. Let's call this total time "t".
t = 2 * (15 m/s * sin(40 degrees)) / 9.8 m/s^2
Now, we can find the horizontal distance traveled using the horizontal component and the total time.
horizontal distance = horizontal component * t
Finally, we have the horizontal distance traveled down the slope.
b) The time in the air is equal to the total time "t" we calculated earlier.
c) To find the velocity with which the skateboarder lands on the slope, we need to find both the horizontal and vertical components of the velocity at that moment.
The horizontal component of the velocity remains constant and equal to the horizontal component of the initial velocity.
The vertical component of the velocity at the moment of landing can be found by multiplying the acceleration due to gravity by the time spent in the air.
vertical component = acceleration due to gravity * t
Now, we can find the magnitude of the velocity with which the skateboarder lands using the Pythagorean theorem:
velocity = sqrt((horizontal component)^2 + (vertical component)^2)
These calculations will give us the answers to all three questions.