Physics · kinematics

Motion in 1D

Position, displacement, velocity and acceleration along a straight line, and the four equations that tie them together.

Position and displacement

Position · x

Location of an object relative to an origin.

Displacement · Δx

Change in position: \( \Delta x = x_2 - x_1 \). A vector: has direction, can be positive, negative or zero.

Distance

Total path length travelled. A scalar: no direction, always positive or zero.

Walk 5 m forward, then 3 m back

distance 0 m displacement 0 m

Distance adds up every metre walked; displacement only cares where you ended up.

Velocity

Average velocity

Displacement over time: \( v_{avg} = \dfrac{\Delta x}{\Delta t} \), in m/s. A vector quantity.

Instantaneous velocity

Velocity at a specific instant: the limit as \( \Delta t \to 0 \), i.e. the slope of the position-time graph.

Speed

The magnitude of velocity: a scalar.

Acceleration

Average acceleration

Change in velocity over time: \( a_{avg} = \dfrac{\Delta v}{\Delta t} \), in m/s². A vector quantity.

Instantaneous acceleration

Acceleration at a specific instant: the slope of the velocity-time graph.

Kinematic equations

For constant acceleration:

\( v = v_0 + at \)
\( x = x_0 + v_0 t + \tfrac{1}{2}at^2 \)
\( v^2 = v_0^2 + 2a(x - x_0) \)
\( x = x_0 + \tfrac{1}{2}(v_0 + v)\,t \)

Where

\( v_0 \) initial velocity
\( v \) final velocity
\( a \) acceleration
\( t \) time
\( x \) position

See it move

v₀ a

Exercises

Answer each question before revealing it.

Exercises · part 1

What is displacement?

Change in position, Δx = x₂ − x₁: a vector quantity

What is the difference between distance and displacement?

Distance is total path length (scalar); displacement is change in position (vector)

If you walk 5 m forward then 3 m backward, what is your displacement?

2 m forward (5 m − 3 m)

What is your total distance travelled?

8 m (5 m + 3 m)

Exercises · part 2

What is average velocity?

Displacement divided by time, v_avg = Δx / Δt

A car travels 100 m in 5 s. What is its average velocity?

v_avg = 100 m / 5 s = 20 m/s

What does the slope of a position-time graph represent?

Instantaneous velocity

What is acceleration?

Rate of change of velocity, a = Δv / Δt

Exercises · part 3

A car accelerates from 0 to 20 m/s in 4 seconds. What is its acceleration?

a = (20 − 0) / 4 = 5 m/s²

What does positive acceleration mean?

Velocity is increasing in the positive direction

Can an object have negative velocity and positive acceleration?

Yes: it is moving backwards but slowing down (try "backwards, slowing" in the simulator)

What is the slope of a velocity-time graph?

Acceleration

Exercises · part 4

Which kinematic equation relates v, v₀, a and t?

v = v₀ + at

A ball drops from rest. After 2 s, what is its velocity? (g = 10 m/s²)

v = 0 + (10)(2) = 20 m/s downward

How far does it fall?

x = ½at² = ½(10)(2²) = 20 m

A car going 30 m/s brakes with a = −5 m/s². How long does it take to stop?

0 = 30 + (−5)t, so t = 6 s

Exercises · part 5

Using v² = v₀² + 2ax, find the stopping distance for the car above.

0² = 30² + 2(−5)x, so x = 900 / 10 = 90 m

What does constant velocity mean?

Acceleration is zero: v = v₀

If a = 0, which equation simplifies to x = v₀t?

x = x₀ + v₀t + ½at² becomes x = x₀ + v₀t when a = 0

Can displacement be zero while distance is not?

Yes, if you return to the starting point (a round trip)

Exercises · part 6

A train accelerates at 2 m/s² for 10 s, starting from rest. Final velocity?

v = 0 + 2(10) = 20 m/s

How far does it travel?

x = ½(2)(10²) = 100 m

Average velocity during this motion?

v_avg = (0 + 20) / 2 = 10 m/s, or x / t = 100 / 10 = 10 m/s

What is free fall?

Motion under gravity only: a = g ≈ 9.8 m/s² downward