A skateboarder is approaching a set of stairs at a speed of 8 m/s. The stairs drop a vertical distance of h=3 meters and take up a horizontal distance d=5 meters. As the skater reaches the edge she can give herself a vertical component of velocity by jumping. Note: this does not change her horizontal velocity.
a.) What vertical velocity should the skater have in order to clear the stairs?
b.) Repeat with V=7m/s, 6 m/s, 5m/s, 4m/s
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To determine the vertical velocity required for the skateboarder to clear the stairs, we can use the principles of projectile motion.
a.) Given:
Initial horizontal velocity (Vx) = 8 m/s
Vertical drop height (h) = 3 m
Horizontal distance (d) = 5 m
To find the vertical velocity (Vy), we can use the following equation:
Vy^2 = u^2 + 2as
Where:
u is the initial vertical velocity (which we need to find)
a is the acceleration due to gravity (-9.8 m/s^2, since it acts downwards)
s is the vertical distance (h)
First, let's calculate the time it takes for the skateboarder to travel the horizontal distance (d) using the equation:
d = Vx * t
Rearranging the equation, we have:
t = d / Vx
t = 5 m / 8 m/s
t ≈ 0.625 s
Now, let's find the initial vertical velocity (u) using the equation:
Vy^2 = u^2 + 2as
Rearranging the equation, we have:
u^2 = Vy^2 - 2as
Substituting values:
u^2 = 0 - 2 * (-9.8 m/s^2) * 3 m
u^2 = 58.8 m^2/s^2
u ≈ √58.8 m/s
u ≈ 7.67 m/s
Therefore, the vertical velocity required for the skateboarder to clear the stairs is approximately 7.67 m/s.
b.) To repeat the process for different initial speeds, we can follow the same steps as above, adjusting the initial horizontal velocity (Vx) for each case.
Case 1: Vx = 7 m/s
Follow the same steps and calculate the time it takes to travel the horizontal distance (d). Then, use the equation Vy^2 = u^2 + 2as to find the initial vertical velocity (u). Repeat the calculations to find the required vertical velocity.
Case 2: Vx = 6 m/s
Repeat the steps as before, adjusting the initial horizontal velocity (Vx) and finding the required vertical velocity.
Case 3: Vx = 5 m/s
Repeat the steps as before, adjusting the initial horizontal velocity (Vx) and finding the required vertical velocity.
Case 4: Vx = 4 m/s
Repeat the steps as before, adjusting the initial horizontal velocity (Vx) and finding the required vertical velocity.
Following this process, you can find the required vertical velocities for each case and determine if the skateboarder will clear the stairs at each speed.