Combining the Velocity-Time Graphs

The total distance travelled is the total area under the graph from the graph, the shape is that of a trapezium.

 ∴ Total distance covered = Area of a trapezium

=   \( \frac{1}{2}\)(Sum of parallel sides) x h

=   \( \frac{1}{2} \scriptsize (a \: + \: b) \: \times \: h\)

Also, breaking the graph into components, S_1, S_2, S₃

S₁ is a right angle triangle = distance covered = Area of a right angle triangle

S₁ = \( \frac{1}{2} \scriptsize \: \times \: base \: \times \: height \\ = \frac{1}{2} \scriptsize bh\) 

S₂ = Area of a rectangle = total distance covered = L x b

S₃ = Area of a right angle triangle = \( \frac{1}{2} \scriptsize \: \times \: base \: \: \times \: height \\ =\frac{1}{2} \scriptsize bh \)

  ∴ the total distance covered = S₁ +  S₂ + S₃

= \(\normalsize \frac{1}{2} \scriptsize bh \: + \: (L \: \times \: b) \: + \: \normalsize \frac{1}{2} \scriptsize bh \)

Example 5.4.1:

A car is travelling with an initial velocity of 30 ms^-1 and attains a final velocity of 80 ms^-1 in 20 seconds. Find the acceleration of the car.