A particle is moving in a straight line with velocity $v$ m s$^{-1}$ at time $t$ seconds as shown by the graph below - AQA - A-Level Maths: Mechanics - Question 15 - 2020 - Paper 2

To apply the trapezium rule, we first identify the interval and the corresponding $y$-values based on the graph:

• We divide the interval [$20, 100$] into four equal strips, giving us a strip width of: $h = \frac{100 - 20}{4} = 20$

• The $x$-values (time) are: $20, 40, 60, 80, 100$.

• The corresponding $y$-values (velocity) at these points are:

  • $y_0 = 120$ (at $t=20$)
  • $y_1 = 140$ (at $t=40$)
  • $y_2 = 80$ (at $t=60$)
  • $y_3 = 20$ (at $t=80$)
  • $y_4 = 0$ (at $t=100$)

Using the trapezium rule formula:

$\text{Area} \approx \frac{h}{2} (y_0 + 2y_1 + 2y_2 + 2y_3 + y_4)$

Substituting the values:

$\text{Area} \approx \frac{20}{2} (120 + 2(140) + 2(80) + 2(20) + 0)$ $= 10 (120 + 280 + 160 + 40)$ $= 10 \times 600 = 6000$

Thus, the estimated distance travelled by the particle from $20$ s to $100$ s is $6000$ meters.