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:

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