Motion in a Plane

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**Displacement**: Vector quantity representing the shortest distance between initial and final positions with direction. **Path length**: Scalar quantity representing the actual distance traveled. **

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**Average velocity** is the displacement divided by time interval. **Formula**: v_av = (r₂ - r₁)/(t₂ - t₁) = Δr/Δt In component form: v_av = ((x₂-x₁)/(t₂-t₁))î + ((y₂-y₁)/(t₂-t₁))ĵ **Note**: It's a

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**Instantaneous velocity** is the velocity of an object at a specific instant of time. **Formula**: v = lim(Δt→0) Δr/Δt = dr/dt In components: v = (dx/dt)î + (dy/dt)ĵ **Direction**: Always tangent

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For uniform acceleration, the three equations are: 1. **v = u + at** (velocity-time relation) 2. **s = ut + ½at²** (displacement-time relation) 3. **v² = u² + 2as** (velocity-displacement relation)

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**Given**: r = (5t²î + 3tĵ) m **Solution**: Velocity: v = dr/dt = (10tî + 3ĵ) m/s At t = 2s: v = (20î + 3ĵ) m/s |v| = √(20² + 3²) = √409 = 20.22 m/s Acceleration: a = dv/dt = 10î m/s² |a| = 10 m/s²

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**Relative velocity** of object A with respect to object B is the velocity that A appears to have when observed from B. **Formula**: v_AB = v_A - v_B **Example**: Train A moves at 80 km/h east, Trai

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**Given**: v_plane = 300ĵ km/h, v_wind = 100î km/h **Solution**: Velocity relative to ground = v_plane + v_wind v_resultant = 100î + 300ĵ km/h **Magnitude**: |v| = √(100² + 300²) = √100000 = 316.2 k

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**Projectile motion** is the motion of an object thrown with some initial velocity under the influence of gravity alone. **Key characteristics**: 1. **Horizontal motion**: Uniform (no acceleration, v