The drawing shows a skateboarder moving at 7.59 m/s along a horizontal section of a track that is slanted upward by 55.0 ° above the horizontal at its end, which is 0.741 m above the ground. When she leaves the track, she follows the characteristic path of projectile motion. Ignoring friction and air resistance, find the maximum height H to which she rises above the end of the track.
Hmmm. The way I see it, kinetic energy is lost during her rise up the ramp. So, at the moment of takeoff, her KE is
1/2 mv^2 - 1/2 mgh
= m(1/2 * 7.59^2 - 1/2 * 9.8 * 0.741)
= 25.17m
Now, that means that her speed is then 7.10 m/s
The vertical component of that is 5.81 m/s, so counting from the end of the track,
H(t) = 5.81t - 4.9t^2
which has a maximum at (.59,1.72)
To find the maximum height (H) to which the skateboarder rises above the end of the track, we can use the principles of projectile motion.
1. Start by breaking down the skateboarder's motion into horizontal and vertical components.
- The horizontal component, Vx, remains constant throughout the motion since there is no horizontal force acting on the skateboarder.
- The vertical component, Vy, changes due to the acceleration of gravity.
2. Determine the initial velocity components:
- The initial velocity in the horizontal direction, Vx, can be found using the skateboarder's speed along the track (7.59 m/s) and the angle of the track (55.0 °).
Vx = speed * cos(angle)
- Calculate Vx using this formula:
Vx = 7.59 m/s * cos(55.0 °)
- The initial velocity in the vertical direction, Vy, can be found using the skateboarder's speed along the track (7.59 m/s) and the angle of the track (55.0 °).
Vy = speed * sin(angle)
- Calculate Vy using this formula:
Vy = 7.59 m/s * sin(55.0 °)
3. Calculate the time of flight, t:
- The time of flight is the total time the skateboarder is in the air before hitting the ground.
- Since the horizontal component of velocity doesn't change, we can calculate time using the vertical component of velocity.
- Use Vy and the acceleration due to gravity (9.8 m/s^2) to calculate the time of flight:
t = Vy / g
- Calculate t using this formula:
t = (7.59 m/s * sin(55.0 °)) / 9.8 m/s^2
4. Determine the maximum height, H:
- The maximum height reached by the skateboarder can be calculated using the time of flight and the initial vertical velocity.
- Use the formula for vertical displacement with constant acceleration:
H = (Vy^2) / (2g)
- Calculate H using this formula:
H = ((7.59 m/s * sin(55.0 °))^2) / (2 * 9.8 m/s^2)
Plug in the values into the formulas and calculate the maximum height (H).