Section 1.1: Position, Velocity & Acceleration (1D)
In one-dimensional motion, objects move along a straight line. We describe motion using position, velocity, and acceleration.
Key Concepts:
- Position (x): Location along a line.
- Displacement (Δx): Change in position, Δx = x₂ - x₁.
- Velocity (v): Rate of change of position, v = dx/dt.
- Acceleration (a): Rate of change of velocity, a = dv/dt.
Equations of Motion (Constant Acceleration)
For constant acceleration:
- \(v = v_0 + a t\)
- \(x = x_0 + v_0 t + \frac{1}{2} a t^2\)
- \(v^2 = v_0^2 + 2 a (x - x_0)\)
Example 1
A car starts from rest and accelerates at 2 m/s² for 5 seconds. Find its final velocity and displacement.
Final velocity: \(v = v_0 + a t = 0 + 2(5) = 10 \text{ m/s}\)
Displacement: \(x = x_0 + v_0 t + \frac{1}{2} a t^2 = 0 + 0 + 0.5(2)(5^2) = 25 \text{ m}\)
Practice Problems
- A bike moves with constant acceleration 1.5 m/s² from rest for 8 seconds. Find velocity and displacement.
- Object has velocity 20 m/s and decelerates at 2 m/s². How long until it stops? How far does it travel?
- A runner covers 100 m in 10 s starting from rest. Find acceleration assuming constant.
- An object moves along x-axis: x = 5 + 3t + t². Find velocity and acceleration at t = 2 s.