2D Motions and Projectile Motions

2D and 3D Position, Velocity, and Acceleration

  • Position vectors-

    • 2D: \mathbf{r} = x\hat{\mathbf{i}} + y\hat{\mathbf{j}}.

    • 3D: \mathbf{r} = x\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}}.

  • Velocity: Rate of change of position-

    • 2D/3D: \mathbf{v} = \frac{d\mathbf{r}}{dt} = \dot{x}\hat{\mathbf{i}} + \dot{y}\hat{\mathbf{j}} \;(\text{and } +\dot{z}\hat{\mathbf{k}} \text{ in 3D}).

  • Acceleration: Rate of change of velocity-

    • 2D/3D: \mathbf{a} = \frac{d\mathbf{v}}{dt} = \ddot{x}\hat{\mathbf{i}} + \ddot{y}\hat{\mathbf{j}} \;(\text{and } +\ddot{z}\hat{\mathbf{k}} \text{ in 3D}).

  • Key relations-

    • Velocity is d\mathbf{r}/dt.

    • Acceleration is d\mathbf{v}/dt.

Speed and Components

  • Speed (magnitude of velocity)-

    • 2D: ||\mathbf{v}|| = \sqrt{\dot{x}^2 + \dot{y}^2}.

    • 3D: ||\mathbf{v}|| = \sqrt{\dot{x}^2 + \dot{y}^2 + \dot{z}^2}.

  • Velocity components: vx = \dot{x}, \quad vy = \dot{y}, \quad v_z = \dot{z}.

  • Acceleration components: ax = \ddot{x}, \quad ay = \ddot{y}, \quad a_z = \ddot{z}.

Direction and Acceleration

  • Acceleration can exist even with constant speed if velocity direction changes (e.g., circular motion).

Independence and Superposition in Multi-Dimensional Motion

  • Orthogonal components (x, y, z) are independent; time links them.

  • Solve by superposing 1D motions.

  • Practical takeaway: Break problems into independent 1D motions, then combine results via time.

Projectile Motion Overview

  • Projectile motion: Free fall with gravity as the only force, so acceleration is only in the y-direction.

  • Acceleration components: x-direction acceleration is zero; y-direction acceleration is due to gravity (approx. 9.8 \text{ m s}^{-2} downwards).

  • Consequence: x-motion has constant horizontal velocity.

Kinematics in x

  • Without acceleration in x, horizontal position changes linearly with time at a constant velocity.

Kinematics in y

  • Vertical motion is under constant downward acceleration due to gravity, influencing vertical position and velocity.

Independence of axes; common time

  • x and y motions are independent but share the same time parameter.

  • Time found from y-motion can be used for x-motion.

Superposition and parabolic trajectory

  • Motion is a superposition of constant-velocity horizontal motion and vertical free fall, resulting in a parabolic path.

  • The path remains parabolic even from a moving frame.

  • On level ground, the trajectory is symmetric.

Time of flight and height

  • Time of flight depends solely on vertical motion.

  • Similar vertical initial conditions result in similar flight times (e.g., dropping from rest vs. horizontal launch from same height).

  • Reaching the same maximum height implies the same time of flight.

  • Higher maximum height generally leads to longer flight time.

Practical problem-solving tips

  • Determine time using vertical equations first, then apply it to horizontal equations.

  • Since horizontal velocity is constant, horizontal range is proportional to the time of flight.