Speed-time and velocity-time graphs - Higher
Speed-time graphs show speed on the vertical axis and time on the horizontal axis. The In a graph, the gradient is the steepness of the line. The greater the gradient, the greater the rate of change. 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.
The area under a speed-time graph represents the distance travelled. Likewise, the area under a velocity-time graph represents the Quantity describing the distance from the start of the journey to the end in a straight line with a described direction, eg 50 km due north of the original position. 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.
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 \( \frac{1}{2} \times {3}\times {8} = 12\:{\text{m}}\)
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