A skier leaves the top of a ski ramp at an angle of 25 degrees with an initial velocity of 10
m/s. The ramp is 60 m high, neglecting air resistance determine:
a.
The time it takes the sk
ier to reach the ground.
b.
How far away from the base does he land?
c.
How high above the
ground
does the skier travel?
To solve this problem, we can break it down into three parts:
a. Determining the time it takes for the skier to reach the ground.
b. Finding out how far away from the base the skier lands.
c. Calculating the maximum height above the ground reached by the skier.
Let's start by solving part a, determining the time it takes the skier to reach the ground:
1. We can resolve the initial velocity of the skier into its vertical and horizontal components using trigonometry.
- The vertical component is calculated as: initial velocity * sin(angle)
vertical component = 10 m/s * sin(25 degrees)
- The horizontal component is calculated as: initial velocity * cos(angle)
horizontal component = 10 m/s * cos(25 degrees)
2. Now we can determine the time it takes for the skier to reach the ground by using the formula:
- time = sqrt(2 * height / g), where g is the acceleration due to gravity (approximately 9.8 m/s^2)
time = sqrt(2 * 60 m / 9.8 m/s^2)
Moving on to part b, finding out how far away from the base the skier lands:
1. We need to calculate the skier's horizontal displacement. We already know the horizontal component of the initial velocity, which is 10 m/s * cos(25 degrees).
2. To find the time of flight (the total time the skier is in the air), we use the formula:
- time of flight = 2 * time
3. Finally, we can calculate the horizontal displacement:
- horizontal displacement = horizontal component * time of flight
Lastly, let's solve part c, calculating the maximum height above the ground reached by the skier:
1. To find the maximum height above the ground, we need to determine the vertical displacement.
- We can use the formula: vertical displacement = vertical component * time of flight + 0.5 * (-g) * (time of flight)^2
Now, let's plug in the values and calculate the answers:
a. The time it takes the skier to reach the ground:
- Calculate the vertical component, vertical component = 10 m/s * sin(25 degrees)
- Calculate the time, time = sqrt(2 * 60 m / 9.8 m/s^2)
b. How far away from the base does the skier land:
- Calculate the horizontal displacement, horizontal displacement = horizontal component * time of flight
c. How high above the ground does the skier travel:
- Calculate the maximum height above the ground, vertical displacement = vertical component * time of flight + 0.5 * (-g) * (time of flight)^2
By following these steps, you will be able to find the solutions to the given problem.