2. Motion Graph Gradients and Areas

Core Graphical Principles

In motion graphs, we frequently use the gradient of a line and the area under a line to find the values of physical quantities.

The Gradient: Calculated by choosing two points on the line and dividing the change in the y-axis by the change in the x-axis.

$$Gradient = \dfrac{\Delta y}{\Delta x}$$

The Area Under the Curve: In IB Physics, you will mostly calculate the area under straight lines by breaking the space into geometric shapes (rectangles and triangles).

Displacement-Time Graphs

A displacement-time graph plots how far an object is from a reference point over time.

Gradient = Velocity

Since the y-axis is displacement ($s$) and the x-axis is time ($t$):

$$Gradient = \dfrac{\Delta s}{\Delta t} = Velocity$$

Interpreting the lines:

  • Horizontal Line: The displacement stays the same. The object is stationary ($v=0$).
  • Straight Sloped Line: The displacement increases by the same amount each second. The object is moving with constant velocity.
  • Curved Line: The displacement covered each second increases. The gradient is changing, meaning the object is accelerating.

Velocity-Time Graphs

A velocity-time graph is highly informative because both its gradient and its enclosed area correspond to kinematic quantities.

Gradient = Acceleration

Since the y-axis is velocity ($v$) and the x-axis is time ($t$):

$$Gradient = \dfrac{\Delta v}{\Delta t} = Acceleration$$

Area = Displacement

Because displacement is velocity multiplied by time ($s = v \times t$), the space bounded between the graph line and the time axis is the displacement.

$$Area \ under \ graph = Displacement$$

Interpreting the lines:

  • Horizontal Line: The velocity stays the same. The object moves with constant velocity ($a=0$).
  • Straight Sloped Line: The velocity increases by the same amount each second. The object undergoes uniform acceleration.
  • Curved Line: The velocity increases by a larger amount each second. The object undergoes non-uniform acceleration.

⚠️ Calculating Bounded Area

Remember to split complex areas under a velocity-time graph into simple shapes:

  • Rectangle Area: $base \times height$
  • Triangle Area: $\frac{1}{2} \times base \times height$

Total Displacement = Area of Rectangle + Area of Triangle.