A car is moving in a straight line along a horizontal road. The graph below shows how the car's velocity $v \, \text{ms}^{-1}$ changes with time, $t$ seconds. Over the period $0 \leq t \leq 15$ the car has a total displacement of $-7$ metres. Initially the car has velocity $0 \, \text{ms}^{-1}$. Find the next time when the velocity of the car is $0 \, \text{ms}^{-1}$.

Photo AI

A car is moving in a straight line along a horizontal road - AQA - A-Level Maths: Pure - Question 15 - 2022 - Paper 2

Question 15

A car is moving in a straight line along a horizontal road. The graph below shows how the car's velocity $v \, \text{ms}^{-1}$ changes with time, $t$ seconds. Over ... show full transcript

Worked Solution & Example Answer:A car is moving in a straight line along a horizontal road - AQA - A-Level Maths: Pure - Question 15 - 2022 - Paper 2

Step 1

Find the area above the time axis

96%

114 rated

Answer

To find the time when the velocity is zero again, we first calculate the area of the triangle above the time axis in the graph. The formula for the area of a triangle is given by:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

In this case, the triangle from $0$ to $10$ seconds has a base of $10$ seconds and a height of $4 \, \text{ms}^{-1}$ , which gives:

$\text{Area}_{\text{above}} = \frac{1}{2} \times 10 \times 4 = 20 \; \text{m}$

Step 2

Find the area below the time axis

99%

104 rated

Answer

Next, we compute the area of the trapezoid below the time axis from $10$ to $15$ seconds. The area can be expressed as:

$\text{Area}_{\text{below}} = 2(10 - t) + 20$

where $t$ is the time. We will evaluate this for the interval from $10$ to $15$ seconds.

Step 3

Set up the equation for displacement

96%

101 rated

Answer

The total displacement is the area above minus the area below. Thus, we set up the equation:

$\text{Total Displacement} = \text{Area}_{\text{above}} - \text{Area}_{\text{below}} = 20 - (2(10 - t) + 20)$

Setting this equal to $-7$ gives:

$20 - (20 - 2t) = -7$

Simplifying the equation:

$2t = 7 \implies t = 3.5 \, \text{seconds}$

Step 4

Find next instance when velocity is 0

98%

120 rated

Answer

We need to consider the period after $10$ seconds. The equation continues from there. By evaluating when the car returns to $0 \, \text{ms}^{-1}$ for the next instance:

From the graph, the velocity appears to hit zero again at: $t = 8.25 \, \text{seconds}$

A car is moving in a straight line along a horizontal road - AQA - A-Level Maths: Pure - Question 15 - 2022 - Paper 2

Worked Solution & Example Answer:A car is moving in a straight line along a horizontal road - AQA - A-Level Maths: Pure - Question 15 - 2022 - Paper 2

Find the area above the time axis

Find the area below the time axis

Set up the equation for displacement

Find next instance when velocity is 0

Join the A-Level students using SimpleStudy...

Other A-Level Maths: Pure topics to explore