Speed-time and velocity-time graphs - Higher

Speed-time graphs show speed on the vertical axis and time on the horizontal axis. The gradient of a speed-time graph represents acceleration because:

gradient =  \frac {\text{change in y}}{\text{change in x}} = \frac {\text{change in speed}}{\text{change in time}}

 = \frac {\text{change in metres per second}}{\text{change in seconds}} = metres per second per second.

Metres per second per second can be written as = {m/}{s}^{2} or = {ms}^{-2}.

A negative gradient shows the rate of 'slowing down' or deceleration.

Velocity-time graphs show velocity on the vertical axis. Acceleration is still represented by the gradient.

curriculum-key-fact
The gradient of speed-time and velocity-time graphs represent acceleration.

The area under a speed-time graph represents the distance travelled. Likewise, the area under a velocity-time graph represents the displacement of the moving object. If the velocity is always positive, then the displacement will be the same as the distance.

Example

Describe what is happening in this journey.

Finding the distance travelled for a journey shown on the velocity-time graph

Between 0 and 4 seconds:

  • the object is accelerating at \frac {8}{4} = 2{\text{~m/s}}^2
  • it travels \frac {1}{2} \times {4} \times {8} = 16~m

Between 4 and 7 seconds:

  • the object is travelling at a constant velocity of 8 m/s
  • it travels {3} \times {8} = 24~m

Between 7 and 10 seconds:

  • the object is accelerating at  \frac {-8} {3} = -2 \frac {2}{3} {\text{~m/s}}^2
  • this means it is slowing down or decelerating at a rate of {2} \frac {2}{3} {\text{~m/s}}^2
  • it travels  \frac{1}{2} \times {3}\times {8} = 12\:{\text{m}}