The drawing shows a skateboarder moving at 5.34 m/s along a horizontal section of a track that is slanted upward by 47.7 ° above the horizontal at its end, which is 0.795 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 whioch she rises above the end of the track.
To find the maximum height to which the skateboarder rises above the end of the track, we can use the principles of projectile motion.
1. Resolve the initial velocity of the skateboarder into its vertical and horizontal components:
- Vertical component: Vyi = V * sin(θ)
- Horizontal component: Vxi = V * cos(θ)
where V is the initial velocity of the skateboarder (5.34 m/s) and θ is the angle of the track (47.7 °).
2. Calculate the time taken for the skateboarder to reach the maximum height using the vertical motion equation:
- Vf = Vi + at
- Vf = 0 (at the maximum height)
- Vi = Vyi
- a = -g (acceleration due to gravity, approximately -9.8 m/s²)
Solving for t:
0 = Vyi - gt
t = Vyi / g
3. Calculate the maximum height reached using the vertical motion equation:
- d = Vit + (1/2)at²
- d = 0.795 m (height above the ground at the end of the track)
- Vi = Vyi
- a = -g
- t = Vyi / g
Solving for H:
H = 0 + Vyi * (Vyi / g) + (1/2) * (-g) * ((Vyi / g)²)
H = (Vyi² / (2g))
4. Substitute the values into the equations to calculate the maximum height:
- Vyi = V * sin(θ)
- g = 9.8 m/s²
H = (V * sin(θ))² / (2 * 9.8)
Let's calculate the maximum height using the given values.
To find the maximum height to which the skateboarder rises above the end of the track, we can use the principles of projectile motion. The skateboarder's initial velocity along the horizontal section of the track can be resolved into two components: horizontal and vertical.
Given:
Initial velocity (v): 5.34 m/s
Angle of the track (θ): 47.7°
Height of the track (h): 0.795 m
1. Resolve the initial velocity into its horizontal and vertical components:
Vertical component (v₁) = v * sin(θ)
Horizontal component (v₂) = v * cos(θ)
2. Calculate the time it takes for the skateboarder to reach the highest point of the projectile motion using the vertical component:
Use the equation: v = u + at, where u is the initial vertical velocity, a is the acceleration, and t is the time.
At the highest point, the vertical velocity is zero (v = 0). Therefore, the equation becomes: 0 = v₁ - gt, where g is the acceleration due to gravity (9.8 m/s²).
Solve for t: t = v₁ / g
3. Calculate the maximum height attained by the skateboarder above the end of the track:
Use the equation for vertical displacement: d = v₁ * t - 0.5 * g * t²
The vertical displacement is equal to the maximum height (H) reached by the skateboarder.
Therefore, H = v₁ * t - 0.5 * g * t²
Plug in the given values and solve the equation to find the maximum height (H).