Solved: The braking distance of a vehicle is defined as the distance travelled from where... [IB Mathematics Applications & Interpretation (AI)]

Question

SLPaper 2

The braking distance of a vehicle is defined as the distance travelled from where the brakesare applied to the point where the vehicle comes to a complete stop.

The speed, $s\,{\text{m}}\,{{\text{s}}^{ - 1}}$ , and braking distance, $d\,{\text{m}}$ , of a truck were recorded. This information issummarized in the following table.

This information was used to create Model A, where $d$ is a function of $s$ , $s$ ≥ 0.

Model A: $d\left( s \right) = p{s^2} + qs$ , where $p$ , $q \in \mathbb{Z}$

At a speed of $6\,{\text{m}}\,{{\text{s}}^{ - 1}}$ , Model A can be represented by the equation $6p + q = 2$ .

Additional data was used to create Model B, a revised model for the braking distance of a truck.

Model B: $d\left( s \right) = 0.95{s^2} - 3.92s$

The actual braking distance at $20\,{\text{m}}\,{{\text{s}}^{ - 1}}$ is $320\,{\text{m}}$ .

Write down a second equation to represent Model A, when the speed is $10\,{\text{m}}\,{{\text{s}}^{ - 1}}$ .

[2]

Verified

Solution

$p{\left( {10} \right)^2} + q\left( {10} \right) = 60$ M1

$10p + q = 6\left( {100p + 10q = 60} \right)$ A1

[2 marks]

Find the values of $p$ and $q$ .

[2]

Verified

Solution

$p = 1$ , $q = - 4$ A1A1

Note: If $p$ and $q$ are both incorrect then award M1A0 for an attemptto solve simultaneous equations.

[2 marks]

Find the coordinates of the vertex of the graph of $y = d\left( s \right)$ .

[2]

Verified

Solution

(2, −4) A1A1

Note: Award A1 for each correct coordinate.
Award A0A1 if parentheses are missing.

[2 marks]

Using the values in the table and your answer to part (b), sketch the graph of $y = d\left( s \right)$ for 0 ≤ $s$ ≤ 10 and −10 ≤ $d$ ≤ 60, clearly showing the vertex.

[3]

Verified

Solution

Note: Award A1 for smooth quadratic curve on labelled axes and within correct window.
Award A1 for the curve passing through (0, 0) and (10, 60). Award A1 for thecurve passing through their vertex. Follow through from part (b).

[3 marks]

Hence, identify why Model A may not be appropriate at lower speeds.

[1]

Verified

Solution

the graph indicates there are negative stopping distances (for low speeds) R1

Note: Award R1 for identifying that a feature of their graph results in negativestopping distances (vertex, range of stopping distances…).

[1 mark]

Use Model B to calculate an estimate for the braking distance at a speed of $20\,{\text{m}}\,{{\text{s}}^{ - 1}}$ .

[2]

Verified

Solution

$0.95 \times {20^2} - 3.92 \times 20$ (M1)

$= 302\,\left( {\text{m}} \right)\,\,\left( {301.6 \ldots } \right)$ A1

[2 marks]

Calculate the percentage error in the estimate in part (e).

[2]

Verified

Solution

$\left| {\frac{{301.6 - 320}}{{320}}} \right| \times 100$ M1

$= 5.75$ (%) A1

[2 marks]

It is found that once a driver realizes the need to stop their vehicle, 1.6 seconds will elapse,on average, before the brakes are engaged. During this reaction time, the vehicle willcontinue to travel at its original speed.

A truck approaches an intersection with speed $s\,{\text{m}}\,{{\text{s}}^{ - 1}}$ . The driver notices the intersection’straffic lights are red and they must stop the vehicle within a distance of $330\,{\text{m}}$ .

Using model B and taking reaction time into account, calculate the maximum possiblespeed of the truck if it is to stop before the intersection.

[3]

Verified

Solution

$330 = 1.6 \times s + 0.95 \times {s^2} - 3.92 \times s$ M1A1

Note: Award M1 for an attempt to find an expression including stoppingdistance (model B) and reaction distance, equated to 330.Award A1 for a completely correct equation.

$19.9\left( {{\text{m}}\,{{\text{s}}^{ - 1}}} \right)\,\,\left( {19.8988 \ldots } \right)$ A1

[3 marks]

Mathematics Applications & Interpretation (AI) International Baccalaureate (IB) Practice Question: The braking distance of a vehicle is defined as the distance travelled from where the brakesare applied to the point whe...