Understanding Projectile Motion
Projectile motion is a form of motion experienced by an object or particle (a projectile) that is projected near the Earth's surface and moves along a curved path under the action of gravity only.
Problem Breakdown
Given values:
- Initial velocity ($u$) = $500 \text{ ms}^{-1}$
- Angle of projection ($\theta$) = $30^{\circ}$
- Acceleration due to gravity ($g$) $\approx 9.8 \text{ ms}^{-2}$
1. Time to Reach Greatest Height ($t$)
The time taken to reach the maximum height is given by the formula: $t = \frac{u \sin \theta}{g}$ $t = \frac{500 \sin(30^{\circ})}{9.8} = \frac{500 \times 0.5}{9.8} = \frac{250}{9.8} \approx 25.51 \text{ s}$
2. Greatest Height ($H$)
The formula for the maximum height attained is: $H = \frac{u^2 \sin^2 \theta}{2g}$ $H = \frac{500^2 \times (\sin 30^{\circ})^2}{2 \times 9.8} = \frac{250000 \times 0.25}{19.6} = \frac{62500}{19.6} \approx 3188.78 \text{ m}$
3. Horizontal Range ($R$)
The horizontal distance covered is determined by: $R = \frac{u^2 \sin(2\theta)}{g}$ $R = \frac{500^2 \times \sin(60^{\circ})}{9.8} = \frac{250000 \times 0.866}{9.8} \approx 22091.84 \text{ m}$
Summary of Results
- Time to max height: $\approx 25.51 \text{ s}$
- Greatest height: $\approx 3188.78 \text{ m}$
- Horizontal range: $\approx 22091.84 \text{ m}$