Section 1.8: Advanced 2D Problems
Advanced two-dimensional problems involve combining projectile motion, relative motion, and vector decomposition. They often require splitting velocities into components and using kinematic equations in both x and y directions.
- Use components: \( x = x_0 + v_x t \), \( y = y_0 + v_y t + \frac{1}{2} a_y t^2 \)
- Combine velocities: \( \vec{v}_{result} = \vec{v}_x + \vec{v}_y \)
- Relative motion may alter initial velocity components
Example 1
A projectile is fired at 20 m/s at 30° above the horizontal. Find its maximum height and range.
Horizontal velocity: \( v_x = 20 \cos 30^\circ \approx 17.32 \text{ m/s} \)
Vertical velocity: \( v_y = 20 \sin 30^\circ = 10 \text{ m/s} \)
Maximum height: \( H = \frac{v_y^2}{2g} = \frac{10^2}{2 \cdot 9.8} \approx 5.10 \text{ m} \)
Time of flight: \( t = \frac{2 v_y}{g} = \frac{20}{9.8} \approx 2.04 \text{ s} \)
Range: \( R = v_x t = 17.32 \cdot 2.04 \approx 35.3 \text{ m} \)
Example 2
A river flows at 4 m/s. A boat can move at 3 m/s in still water. Find the angle to cross directly north.
Boat velocity relative to water: \( v_b = 3 \text{ m/s} \)
River velocity: \( v_r = 4 \text{ m/s} \)
Required heading angle: \( \theta = \tan^{-1}(v_r/v_b) = \tan^{-1}(4/3) \approx 53.13^\circ \) upstream from north
Practice Problems
- A projectile launched at 25 m/s at 45°. Find max height, range, and flight time.
- A swimmer crosses a 50 m wide river with current 2 m/s. If swimmer speed is 3 m/s relative to water, find time to cross and displacement downstream.
- An airplane flies at 80 m/s east with wind 20 m/s north. Determine velocity relative to ground and direction.
- A ball rolls off a 10 m high table at 5 m/s. Find landing distance and time of flight.
- A boat aims to cross a 200 m wide river directly. River flows at 5 m/s; boat speed 10 m/s. Find required heading and time.