Here is a speed-time graph for a car. (a) Work out an estimate for the distance the car travelled in the first 30 seconds. (b) Is your answer to part (a) an underestimate or an overestimate of the actual distance the car travelled in the first 30 seconds? Give a reason for your answer. Julian used the graph to answer this question. Work out an estimate for the acceleration of the car at time 60 seconds. Here is Julian's working. acceleration = speed ÷ time = 13 ÷ 60 = 0.216 m/s² Julian's method does not give a good estimate of the acceleration at time 60 seconds. (c) Explain why.

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Here is a speed-time graph for a car - Edexcel - GCSE Maths - Question 17 - 2020 - Paper 3

Question 17

Here is a speed-time graph for a car. (a) Work out an estimate for the distance the car travelled in the first 30 seconds. (b) Is your answer to part (a) an undere... show full transcript

Worked Solution & Example Answer:Here is a speed-time graph for a car - Edexcel - GCSE Maths - Question 17 - 2020 - Paper 3

Step 1

Work out an estimate for the distance the car travelled in the first 30 seconds.

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Answer

To estimate the distance traveled, we calculate the area under the speed-time graph for the first 30 seconds. The graph appears to be a curve that can be approximated by a trapezoid. First, we can estimate the speed at 0 seconds (0 m/s), at 15 seconds (approximately 12 m/s), and at 30 seconds (approximately 13 m/s). Using the trapezoidal rule, the formula for the area of the trapezoid is:

$A = \frac{1}{2} (b_1 + b_2) h$

where:

$b_1$ is the speed at 0 seconds (0 m/s)
$b_2$ is the speed at 30 seconds (approximately 13 m/s)
$h$ is the time interval (30 seconds)

Substituting the values:

$A = \frac{1}{2} (0 + 13) \times 30 = \frac{13 \times 30}{2} = 195 ext{ m}$

Thus, the estimated distance traveled in the first 30 seconds is approximately 195 meters.

Step 2

Is your answer to part (a) an underestimate or an overestimate of the actual distance the car travelled in the first 30 seconds? Give a reason for your answer.

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Answer

The answer to part (a) is an underestimate of the actual distance the car traveled. This is because the speed-time graph starts at 0 m/s and gradually increases, with a peak speed reached before 30 seconds. The area under the curve (actual distance) is larger than the area of the triangle and trapezoid used in the estimate, since parts of the curve that represent higher speeds are not fully accounted for in the trapezoidal approximation.

Step 3

Work out an estimate for the acceleration of the car at time 60 seconds.

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Answer

To find the acceleration at time 60 seconds, we need to determine the speed at this time from the graph. At 60 seconds, the speed is approximately 14 m/s. The acceleration formula is:

$a = \frac{\Delta v}{\Delta t}$

Where:

$\Delta v$ is the change in speed
$\Delta t$ is the change in time

However, since we are looking for acceleration at a specific time with no preceding change from the graph, we consider the average speed change over an appropriate period.

Julian incorrectly used 60 seconds as the time 't', which is not valid in this context as it does not reflect any previous acceleration or a change. Hence, a better method would involve considering the speed changes over an interval around 60 seconds, rather than simply calculating as Julian did.

Step 4

Explain why.

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Answer

Julian's method does not give a good estimate of the acceleration at time 60 seconds because he has not considered the actual change in speed over time. Acceleration should ideally represent the change in velocity over a specified time interval. Simply dividing speed by time does not account for the curve’s behavior around that time, leading to inaccuracies in the estimate.

Here is a speed-time graph for a car - Edexcel - GCSE Maths - Question 17 - 2020 - Paper 3

Worked Solution & Example Answer:Here is a speed-time graph for a car - Edexcel - GCSE Maths - Question 17 - 2020 - Paper 3

Work out an estimate for the distance the car travelled in the first 30 seconds.

Is your answer to part (a) an underestimate or an overestimate of the actual distance the car travelled in the first 30 seconds? Give a reason for your answer.

Work out an estimate for the acceleration of the car at time 60 seconds.

Explain why.

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