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:
To determine the following, we need to break it down into different components:
1. The horizontal component of the skier's initial velocity.
2. The vertical component of the skier's initial velocity.
3. The time it takes for the skier to reach the ground.
4. The horizontal distance the skier covers during this time.
1. Horizontal component of initial velocity:
The horizontal component of velocity remains constant throughout the motion and is not affected by gravity or the ramp's height. Therefore, the horizontal component of the initial velocity is given by:
Horizontal component = Initial velocity * cos(angle)
Horizontal component = 10 m/s * cos(25 degrees)
2. Vertical component of the initial velocity:
The vertical component of the initial velocity is given by:
Vertical component = Initial velocity * sin(angle)
Vertical component = 10 m/s * sin(25 degrees)
3. Time of flight:
The time it takes for the skier to reach the ground can be determined using the vertical motion equation:
Vertical displacement = Vertical component * time - (1/2) * acceleration due to gravity * time^2
Since the skier starts from rest vertically, the vertical displacement is equal to the ramp's height.
60 m = (10 m/s * sin(25 degrees)) * time - (1/2) * 9.8 m/s^2 * time^2
Solving this equation will give us the time of flight.
4. Horizontal distance covered:
The horizontal distance covered can be determined using the horizontal component of velocity and the time of flight:
Horizontal distance = Horizontal component * time
By solving these equations, we can find the answers:
- Horizontal component of initial velocity:
Horizontal component = 10 m/s * cos(25 degrees)
- Vertical component of initial velocity:
Vertical component = 10 m/s * sin(25 degrees)
- Time of flight:
Solve the equation: 60 m = (10 m/s * sin(25 degrees)) * time - (1/2) * 9.8 m/s^2 * time^2 to find the time.
- Horizontal distance covered:
Horizontal distance = Horizontal component * time
Plug in the values into these equations to find the respective answers.