Motion in a Plane
Easy Overview
Imagine throwing a ball to your friend — it curves through the air in a beautiful arc. That's projectile motion. Now think of a fan's blades spinning around — that's circular motion. This chapter is all about motion that isn't just in a straight line. It's where physics gets two-dimensional and a lot more interesting.
Motion in 2D — Position and Velocity Vectors
In 1D, position was just x(t). In 2D, it's r(t) = x(t)î + y(t)ĵ. Velocity is dr/dt, acceleration is dv/dt. Each component behaves independently — that's the golden rule. The x-motion doesn't care what the y-motion is doing. It's like watching a video — the horizontal and vertical are on separate tracks but play together.
Projectile Motion — The Art of Throwing
A projectile is anything launched into the air and left to gravity. The horizontal velocity is constant (no force horizontally, ignoring air). Vertical velocity changes by g = 9.8 m/s² each second. The path is a parabola. Time of flight = 2u sinθ / g. Maximum height = u² sin²θ / 2g. Range = u² sin2θ / g. The magic angle for max range? 45°.
Uniform Circular Motion
When something moves in a circle at constant speed, its direction keeps changing — so it's accelerating! That's centripetal acceleration, always pointing toward the center. a = v²/r = ω²r. The force causing this is centripetal force, but it's not a separate force — it's tension, gravity, friction, or whatever is keeping the object on the circular path.
Angular Variables — θ, ω, α
Angular displacement θ (in radians), angular velocity ω = dθ/dt (rad/s), angular acceleration α = dω/dt (rad/s²). The relationships are exactly like linear motion: ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ. Just replace x with θ, v with ω, a with α. It's the same math, different letters.
Relative Velocity in 2D
Relative velocity in 2D is just vector subtraction. If you're in a car moving at 20 m/s east and another car goes 15 m/s north, their velocity relative to you is the vector difference. v_AB = v_A − v_B. Rain-man problems, river-boat problems — all just vector addition and subtraction. Draw the triangle, solve with Pythagoras.
Key Points
- •Horizontal and vertical motions are independent of each other
- •Time of flight = 2u sinθ / g
- •Maximum horizontal range at θ = 45°
- •Centripetal acceleration: a = v²/r (always toward center)
- •Angular velocity ω = v/r, period T = 2πr/v
- •Banking of roads provides centripetal force through angled normal reaction
Practice Questions
- A ball is thrown at 30 m/s at 60° to horizontal. Find time of flight, max height, and range.
- Derive an expression for centripetal acceleration.
- A stone tied to a string rotates in a vertical circle. Why does tension vary at different points?
- In a river-boat problem, if river flows at 4 km/h and boat speed is 5 km/h, find the angle to cross directly to the opposite bank.