Section 1.7: Relative Motion
Relative motion describes the velocity of an object as observed from a particular reference frame. Motion is always measured relative to something else.
- Velocity of A relative to B: \( \vec{v}_{A/B} = \vec{v}_A - \vec{v}_B \)
- Velocity of B relative to A: \( \vec{v}_{B/A} = \vec{v}_B - \vec{v}_A = -\vec{v}_{A/B} \)
- Relative motion depends on the observer's frame.
- Horizontal and vertical components are treated independently.
- Used extensively in moving vehicles, river crossing, and air navigation problems.
Example 1
A boat moves at 5 m/s relative to water across a river flowing at 3 m/s. Find the velocity of the boat relative to the riverbank.
Velocity relative to bank: \( v = \sqrt{5^2 + 3^2} = \sqrt{34} \approx 5.83 \text{ m/s} \)
Direction: \( \theta = \tan^{-1}(3/5) \approx 31^\circ \) downstream from the perpendicular
Example 2
A car moves east at 20 m/s, while another moves north at 15 m/s. Find the velocity of the second car relative to the first.
\( \vec{v}_{B/A} = \vec{v}_B - \vec{v}_A = -20 \hat{i} + 15 \hat{j} \)
Magnitude: \( \sqrt{(-20)^2 + 15^2} = \sqrt{625} = 25 \text{ m/s} \)
Direction: \( \theta = \tan^{-1}(15/20) \approx 36.87^\circ \) north of west
Practice Problems
- A swimmer moves 2 m/s relative to water; river flows at 1 m/s. Find speed relative to shore.
- Airplane flies 100 m/s east; wind blows 20 m/s north. Find plane's velocity relative to ground.
- Two cars move at 30 m/s and 40 m/s perpendicular to each other. Determine relative velocity.
- A boat crosses a river 100 m wide flowing at 3 m/s. If boat speed in still water is 4 m/s, find time to cross.