The drawing shows a skateboarder moving at 5.55 m/s along a horizontal section of a track that is slanted upward by 40.9 ° above the horizontal at its end, which is 0.724 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 H to which the skateboarder rises above the end of the track, we can follow these steps:
Step 1: Identify the given information:
- Initial velocity of the skateboarder along the horizontal section of the track (v₀) = 5.55 m/s
- Angle of the upward slanted track (θ) = 40.9°
- Height of the end of the track above the ground (h) = 0.724 m
Step 2: Split the motion into horizontal and vertical components:
Since there is no acceleration in the horizontal direction, the horizontal velocity (v_x) remains constant throughout the motion.
In the vertical direction, we have an initial vertical velocity (v₀y) and acceleration due to gravity (g = 9.8 m/s²).
Step 3: Determine the horizontal and vertical velocities:
To find the vertical component of the initial velocity (v₀y), we can use trigonometry:
v₀y = v₀ * sin(θ)
v₀y = 5.55 m/s * sin(40.9°)
To determine the horizontal component of the initial velocity (v₀x), we can use trigonometry:
v₀x = v₀ * cos(θ)
v₀x = 5.55 m/s * cos(40.9°)
Step 4: Calculate the time taken to reach maximum height:
At the maximum height, the vertical velocity becomes zero. We can use the equation of motion to determine the time taken (t) to reach this point:
v_y = v₀y - g * t = 0
t = v₀y / g
Step 5: Calculate the maximum height reached:
To find the maximum height (H), we can use the equation of motion in the vertical direction:
H = h + v₀y * t - (1/2) * g * t²
H = h + v₀y * (v₀y / g) - (1/2) * g * (v₀y / g)²
Now, substitute the known values and calculate H:
H = 0.724 m + (5.55 m/s * sin(40.9°)) * [(5.55 m/s * sin(40.9°)) / 9.8 m/s²] - (0.5 * 9.8 m/s²) * [(5.55 m/s * sin(40.9°)) / 9.8 m/s²]²
After substituting the values and evaluating the expression, you will obtain the maximum height H to which she rises above the end of the track.