A particle is moving in a straight line with velocity $v$ m s$^{-1}$ at time $t$ seconds as shown by the graph below. Use the trapezium rule with four strips to estimate the distance travelled by the particle during the time period $20 \leq t \leq 100$. Over the same time period, the curve can be very closely modelled by a particular quadratic. Explain how you could find an alternative estimate using this quadratic.

A-Level AQA Maths Pure Further Integration: A particle is moving in a straig

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 Pure - Question 15 - 2020 - Paper 2

Question 15

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A particle is moving in a straight line with velocity $v$ m s$^{-1}$ at time $t$ seconds as shown by the graph below. Use the trapezium rule with four strips to est... show full transcript

Worked Solution & Example Answer: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 Pure - Question 15 - 2020 - Paper 2

Step 1

Use the trapezium rule with four strips

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Answer

To calculate the distance travelled using the trapezium rule, we first determine the width of each strip. The time interval given is from 20 s to 100 s, which gives us a total width of:

$h = \frac{100 - 20}{4} = 20$

Next, we need five points (y-values) corresponding to the time intervals at 20, 40, 60, 80, and 100 seconds:

At $t=20$ , $v=131$ m/s
At $t=40$ , $v=140$ m/s
At $t=60$ , $v=120$ m/s
At $t=80$ , $v=80$ m/s
At $t=100$ , $v=0$ m/s

Now, we apply the trapezium rule formula:

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

Substituting the values we have:

$\text{Area} = \frac{20}{2} \left( 131 + 2(140) + 2(120) + 2(80) + 0 \right)$

Calculating this gives:

First, compute the sums: $131 + 280 + 240 + 160 + 0 = 811$
Then, area becomes: $10 \cdot 811 = 8110$

Thus, the distance travelled is estimated as 8110 m.

Step 2

Explain how you could find an alternative estimate using this quadratic

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Answer

To find an alternative estimate, we can integrate the quadratic equation that models the velocity over the interval from 20 to 100 seconds. The integral represents the area under the curve which is equivalent to the distance travelled.

The integral can be formulated as:

$\text{Distance} = \int_{20}^{100} v(t) \, dt$

By evaluating this integral, we would receive a refined estimate of the distance travelled by the particle, potentially yielding a result closer to the actual value as it accounts for the precise shape of the velocity function during that time period.

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 Pure - Question 15 - 2020 - Paper 2

Worked Solution & Example Answer: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 Pure - Question 15 - 2020 - Paper 2

Use the trapezium rule with four strips

Explain how you could find an alternative estimate using this quadratic

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