1.3 Kinematic Relationships in Two Dimensions

Independence of components

In 2D motion, horizontal and vertical components of position, velocity, and acceleration are independent of each other. Each component obeys the same calculus relationships as 1D motion: velocity is the derivative of position, and acceleration is the derivative of velocity, applied separately to x and y. Gravity only affects the vertical component in projectile motion.

Vectors via calculus

Position, velocity, and acceleration are vector functions of time: r(t) = x(t)î + y(t)ĵ. Velocity v(t) = dr/dt, and acceleration a(t) = dv/dt. To find displacement or velocity from acceleration, integrate each component separately, using initial conditions as constants of integration.

Projectile motion example

A classic case: horizontal velocity stays constant (a_x = 0), while vertical velocity changes due to gravity (a_y = -g). The trajectory's shape (parabola) comes from combining x(t) = x₀ + v₀ₓt with y(t) = y₀ + v₀ᵥt - ½gt². Speed and direction combine both components using Pythagorean theorem and arctangent.

Key terms

Component independence

In 2D motion, x and y components of position, velocity, and acceleration can be analyzed separately using 1D kinematics equations.

Position vector r(t)

r(t) = x(t)î + y(t)ĵ, describing an object's location in 2D as a function of time.

Velocity from position (2D)

v(t) = dr/dt = (dx/dt)î + (dy/dt)ĵ — differentiate each component separately.

Acceleration from velocity (2D)

a(t) = dv/dt = (dvₓ/dt)î + (dvᵥ/dt)ĵ — differentiate each velocity component separately.

Finding magnitude & direction of a vector

Magnitude: √(x² + y²) using Pythagorean theorem. Direction: θ = arctan(y/x).

Try an AP-style question

A projectile is launched at 20 m/s at 30° above level ground. What is its time of flight? (g ≈ 10 m/s²)

1. 3.5 s
2. 4 s
3. 2 s
4. 1 s