Questions: The maximum height (H) of an object launched with initial velocity (v0) at angle (theta) is given by the following formula. [ H=fracv0^2 sin ^2 theta2 g ] If (g=9.8 , m / s^2), find the height of an object when (v0=100 , m / s) and (theta=frac4 pi7 , rad). Do not round any intermediate computations. Round your answer to the nearest hundredth: [ H= , m ]

Solution Steps

• Initial velocity \( v_0 = 100 \, \text{m/s} \)
• Angle \( \theta = \frac{4\pi}{7} \, \text{rad} \)
• Acceleration due to gravity \( g = 9.8 \, \text{m/s}^2 \)

\[ H = \frac{v_0^2 \sin^2 \theta}{2g} \] \[ H = \frac{(100)^2 \sin^2 \left( \frac{4\pi}{7} \right)}{2 \times 9.8} \]

\[ \sin \left( \frac{4\pi}{7} \right) \] Using a calculator, find: \[ \sin \left( \frac{4\pi}{7} \right) \approx 0.781831 \]

\[ \sin^2 \left( \frac{4\pi}{7} \right) \approx (0.781831)^2 \approx 0.611256 \]

\[ H = \frac{100^2 \times 0.611256}{2 \times 9.8} \]

\[ H = \frac{10000 \times 0.611256}{19.6} \] \[ H = \frac{6112.56}{19.6} \approx 311.86 \]

\[ H \approx 311.86 \, \text{m} \]

Final Answer