Unit 5 - Problem 7 ==> Swing Set

Given:

• $$ l_{swing} = 4 $$
• $$ h_{swing} = 1 $$
• $$ m_{person} = 60 $$
• $$ {PE}_{max}  \mbox{is at} 45^\circ   \mbox{back} $$
• $$ \mbox{Jump off is at } 30^\circ  \mbox{forward} $$

Overview of Approach:

1. Consider energy available
2. Convert PE to KE and calculating x and y vectors
3. Calculate total time in air
4. Calculate horizontal distance from take-off

Rationale:

• Considering energy available
• $$ h_{PE_{max}} = l_{swing} \cos45^\circ $$
• $$ PE_{max} = m * g *  h_{PE_{max}} = m * g * l_{swing} \cos45^\circ $$
• $$ h_{jump} = ( l_{swing} + h_{swing} ) - ( l_{swing} \cos30^\circ ) $$
• $$ PE_{jump} = m * g * h_{jump} = m * g * ( ( l_{swing} + h_{swing} ) - ( l_{swing} \cos30^\circ )) $$
• $$ PE_{take-off} = PE_{max} - PE_{jump}$$
• Converting PE to KE and calculating x and y vectors
• $$ KE_{take-off} = PE_{take-off} $$
• $$ \frac{1}{2} m V_0^2 = PE_{take-off} $$
• $$ V_0^2 = \frac{2 PE_{take-off}}{m} $$
• $$ V_0 = \sqrt \frac{2 PE_{take-off}}{m} $$
• $$ V_{0-x} = V_0 \cos30^\circ $$
• $$ V_{0-y} = V_0 \sin30^\circ $$
• Calculate total time in air
• $$ t_{up} = -\frac{(V - V_{0-y})}{a} $$
• $$ t_{up} = -\frac{-V_{0-y}}{g} = \frac{V_{0-y}}{g}$$
• $$ \Delta{y_{up}} = t_{up} \frac{V + V_{0-y}}{2} = t_{up} \frac{V_{0-y}}{2} $$
• $$ \Delta{y_{down}} =  \Delta{y_{up}} + h_{jump} $$
• $$ \Delta{y_{down}} = V t + \frac{1}{2} a t_{down}^2 = \frac{1}{2} g t_{down}^2 $$
• $$ t_{down}^2 = \frac{2 \Delta{y_{down}}}{g}   $$
• $$ t_{down} = \sqrt \frac{2 \Delta{y_{down}}}{g} $$
• $$ t_{total} = t_{up} + t_{down} $$
• Calculate horizontal distance from take-off
• $$ \Delta{x} = V_{0-x} * t_{total} $$

Calculations:

• $$ \cos45^\circ = .707107 $$
• $$ \cos30^\circ = .866025 $$
• $$ \sin30^\circ = .500000 $$
• $$ \Delta{x} = 2.35  \mbox{meters} $$