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Lesson 1: One-Dimensional Kinematics

What You'll Learn

In this lesson you'll study motion along a straight line — displacement, velocity, acceleration — and the kinematic equations that relate them under constant acceleration.
Definition

Displacement

Displacement (Δx\Delta x) is the change in position of an object. It is a vector quantity (has magnitude and direction).

Δx=xf−xi\Delta x = x_f - x_i

Displacement can be positive (forward / right) or negative (backward / left). It differs from distance, which is the total path length traveled (always positive).
Definition

Velocity vs. Speed

Average velocity is displacement divided by elapsed time:

vˉ=ΔxΔt=xf−xitf−ti\bar{v} = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}

Velocity is a vector (has direction). Speed is the magnitude of velocity (always positive).

Instantaneous velocity is the velocity at a single moment — the limit of average velocity as Δt→0\Delta t \to 0.
t (s)x (m)fast (steep)slow (shallow)at restslope = velocity
On a position-vs-time graph the slope of the line equals the velocity. A steeper line means greater speed. A horizontal line means the object is at rest.
Definition

Acceleration

Average acceleration is the rate of change of velocity:

aˉ=ΔvΔt=vf−vitf−ti\bar{a} = \frac{\Delta v}{\Delta t} = \frac{v_f - v_i}{t_f - t_i}

Acceleration is a vector. When velocity and acceleration have the same sign the object speeds up; when they have opposite signs the object slows down (decelerates).

SI unit: m/s2\text{m/s}^2.
t (s)v (m/s)Area = ΔxΔtΔvslope = a
On a velocity-vs-time graph the slope equals the acceleration and the area under the curve equals the displacement.
When acceleration aa is constant, the following four equations relate position xx, velocity vv, acceleration aa, and time tt:

v=v0+atv = v_0 + at
x=x0+v0t+12at2x = x_0 + v_0 t + \tfrac{1}{2}at^2
v2=v02+2a(x−x0)v^2 = v_0^2 + 2a(x - x_0)
x=x0+12(v0+v)tx = x_0 + \tfrac{1}{2}(v_0 + v)t

Choose the equation that contains the three known quantities and the one unknown.
Four equations, five variables (x, v, v₀, a, t). Each equation leaves out one variable.
Ex

Example — Braking Car

A car traveling at 30 m/s30\text{ m/s} applies the brakes and decelerates uniformly at −5 m/s2-5\text{ m/s}^2. How far does it travel before stopping?
Solution Steps
Definition

Free Fall

Free fall is motion under the influence of gravity alone (ignoring air resistance). Near Earth's surface the acceleration due to gravity is:

g≈9.80 m/s2 (downward)g \approx 9.80\text{ m/s}^2 \text{ (downward)}

All kinematic equations apply with a=−ga = -g (taking up as positive).
Ex

Example — Ball Thrown Upward

A ball is thrown straight up with v0=15 m/sv_0 = 15\text{ m/s}. How high does it rise?
Solution Steps
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Practice Problem
easy
A sprinter accelerates from rest at 3.0 m/s23.0\text{ m/s}^2 for 4.0 s4.0\text{ s}. What is her final velocity?
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Practice Problem
medium
A car accelerates from 10 m/s10\text{ m/s} to 30 m/s30\text{ m/s} over a distance of 200 m200\text{ m}. What is the acceleration?
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Practice Problem
easy
An object is dropped from rest. How far does it fall in 3.0 s3.0\text{ s}? (Use g=9.8 m/s2g = 9.8\text{ m/s}^2.)
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Practice Problem
medium
A ball is thrown straight up at 20 m/s20\text{ m/s}. How long until it returns to the same height? (Use g=10 m/s2g = 10\text{ m/s}^2 for simplicity.)
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