Question

part 2: predicting height from speed
now, you must work backward. you are given a \target speed\ and must calculate the starting height.
problem 5:

• predict: you want the skater to have a speed of 6.0 m/s at the bottom. what height must you drop her from?

○ $h=(6.0*6.0)/(2*9.8)$
○ $h = 36/19.6$
○ predicted height: _ m

• verify: set the ramp to the height you just calculated. release the skater.

○ measured speed (at bottom): _ m/s (should be 6.0 m/s!)
problem 6:

• predict: you want the skater to have a speed of 9.0 m/s at the bottom. what height must you dro her from?

○ calculation:
○ predicted height: _ m

• verify: set the ramp to your calculated height.

○ measured speed (at bottom): _ m/s
problem 7:

• predict: you want the skater to have a speed of 10.0 m/s at the bottom. what height must you her from?

○ calculation:
○ predicted height: _ m

• verify: set the ramp to your calculated height.

○ measured speed (at bottom): _ m/s
problem 8: (challenge)

• predict: you want the skater to have a speed of 3.5 m/s at the bottom. what height must y her from?

○ calculation:
○ predicted height: _ m

• verify: set the ramp to your calculated height.

○ measured speed (at bottom): _ m/s

Explanation:

Step1: Use the conservation of energy formula

According to the conservation of energy, the potential energy \(mgh\) at the top is converted into kinetic energy \(\frac{1}{2}mv^{2}\) at the bottom. So \(mgh=\frac{1}{2}mv^{2}\), and we can cancel out the mass \(m\) (since \(m
eq0\)). The formula simplifies to \(h = \frac{v^{2}}{2g}\), where \(g = 9.8m/s^{2}\) (acceleration due to gravity) and \(v\) is the speed at the bottom.

Step2: Calculate for \(v = 9.0m/s\)

Substitute \(v = 9.0m/s\) into the formula \(h=\frac{v^{2}}{2g}\). So \(h=\frac{9.0\times9.0}{2\times9.8}=\frac{81}{19.6}\approx4.13m\).

Step3: Calculate for \(v = 10.0m/s\)

Substitute \(v = 10.0m/s\) into the formula \(h=\frac{v^{2}}{2g}\). So \(h=\frac{10.0\times10.0}{2\times9.8}=\frac{100}{19.6}\approx5.10m\).

Step4: Calculate for \(v = 3.5m/s\)

Substitute \(v = 3.5m/s\) into the formula \(h=\frac{v^{2}}{2g}\). So \(h=\frac{3.5\times3.5}{2\times9.8}=\frac{12.25}{19.6}=0.625m\).

Answer:

• Problem 6: Predicted Height: \(4.13m\)
• Problem 7: Predicted Height: \(5.10m\)
• Problem 8: Predicted Height: \(0.625m\)