Speed, motion, and graphs - distance-time and velocity-time graphs

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Middle School Physics

Speed, Motion, and Graphs

Motion graphs diagram

Consider your morning. You walked to the kitchen, a car passed your window, and maybe a fan was spinning. All of that is motion. But in physics, we do not just say 'it moved.' We say how far? How fast? In which direction? And how do we show this with data?

That is exactly what this lesson is about.

Distance and Displacement Are Not the Same

Before we talk about speed, we need to clear up two words that people almost always confuse.

Distance

The total path an object covers during its journey. It does not matter which direction you went; distance counts every meter traveled.

Scalar quantity (size only)

Displacement

The straight-line distance from your starting point to your ending point, including direction.

Vector quantity (size and direction)

Example: Walk 8 meters to the right, then turn and walk 8 meters back to the starting point.

Total distance = 16 meters

Displacement = 0 meters (ended up at starting position)

Speed

Speed = Distance ÷ Time

SI unit: meters per second (m/s)

Example: A student runs 200 meters in 40 seconds.

Speed = 200 ÷ 40 = 5 m/s

Average speed = total distance ÷ total time. It gives total distance divided by total time, but does not describe whether the student started fast and slowed down or the other way around.

The measurement of how fast something is going is its speed — your car speedometer shows speed. As you drive, it shows different speeds. To figure out your average speed, you would use the equation given.

Velocity

Velocity = Displacement ÷ Time

SI unit: meters per second (m/s) with direction

Velocity is a vector quantity. You must give the direction, e.g., 12 m/s North, 5 m/s downward.

Example: Two cars both driving at 20 m/s — one heading east, one heading west.

Same speed, different velocities: +20 m/s and -20 m/s.

Velocity will always change when there is a change in the direction of an object, even if that object is moving at a constant speed. A ball being swung in a circle is at constant speed, but its velocity is always changing because its direction is always changing.

Acceleration

a = (v − u) ÷ t

u = initial velocity, v = final velocity, t = time taken

SI unit: meters per second squared (m/s²)

Example: A bus starts from rest and reaches 18 m/s in 6 seconds.

a = (18 − 0) ÷ 6 = 3 m/s²

The bus gains 3 m/s of velocity every second.

One important point many students overlook is that acceleration is not just about increasing or decreasing speed. An object moving in a circle at constant speed is still accelerating because its direction is constantly changing, so velocity is constantly changing.

Acceleration is about changing velocity, not just changing speed.

Distance-time Graphs

A distance-time graph shows how distance from a starting point changes with time.

  • Y-axis: Distance
  • X-axis: Time
Graph Feature What It Means
Horizontal line Object is stationary (not moving)
Straight diagonal line Constant speed (same distance in same time)
Steeper slope Higher speed
Curve getting steeper Speeding up (accelerating)
Curve getting flatter Slowing down (decelerating)

Slope of a distance-time graph = Speed

Speed = change in distance ÷ change in time = rise ÷ run

Velocity-time Graphs

Time is on the horizontal axis, velocity on the vertical axis.

  • Horizontal line: Constant velocity (no acceleration)
  • Straight line tilted up: Constant positive acceleration
  • Straight line tilted down: Constant negative acceleration (deceleration)

Slope of a velocity-time graph = Acceleration

Acceleration = change in velocity ÷ change in time

Area under a velocity-time graph = Distance travelled

  • For constant velocity: area = rectangle = time × velocity
  • For constant acceleration from rest: area = triangle = ½ × time × velocity

Velocity-time graphs are very useful because both acceleration and distance can be found from one graph.

Uniform Motion

Uniform motion means the object is moving with constant velocity — same speed and direction.

  • Distance-time graph: straight line at an angle
  • Velocity-time graph: horizontal straight line
  • No net force acting on the object

Non-uniform Motion

Non-uniform motion is when velocity changes — speeding up, slowing down, or changing direction.

  • Distance-time graph: curved line
  • Velocity-time graph: sloped or changing line

Almost everything in the real world is non-uniform motion: a car in traffic, a ball thrown upward, a biker going up a hill.

Non-uniform motion requires a force applied to the object. This causes changes to the energy present in the object. A lot of kinetic energy is associated with the motion of the object.

In order to keep the object in motion, some energy must be lost — sometimes as heat, sometimes as a drop in the object's energy. The relationship between energy and an object in motion is very important to understand.