The graph below models the velocity of a small train as it moves on a straight track for 20 seconds. The front of the train is at the point A when t = 0. The mass of the train is 800kg. The total distance travelled in the 20 seconds. Find the distance of the front of the train from the point A at the end of the 20 seconds. Find the maximum magnitude of the resultant force acting on the train. Explain why, in reality, the graph may not be an accurate model of the motion of the train.

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The graph below models the velocity of a small train as it moves on a straight track for 20 seconds - AQA - A-Level Maths: Pure - Question 14 - 2017 - Paper 2

Question 14

The graph below models the velocity of a small train as it moves on a straight track for 20 seconds. The front of the train is at the point A when t = 0. The mass ... show full transcript

Worked Solution & Example Answer:The graph below models the velocity of a small train as it moves on a straight track for 20 seconds - AQA - A-Level Maths: Pure - Question 14 - 2017 - Paper 2

Step 1

Find the total distance travelled in the 20 seconds.

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Answer

To find the total distance travelled, we need to calculate the area under the velocity-time graph.

Calculate the area for each segment of the graph:

From 0 to 6 seconds, the velocity is 8 m/s:
$ext{Area}_1 = ext{Base} \times \text{Height} = 6 \times 8 = 48 \text{ m}$
From 6 to 10 seconds, the velocity decreases linearly to 0 m/s. The area is a triangle:
$ext{Area}_2 = \frac{1}{2} \times 4 \times 8 = 16 \text{ m}$
From 10 to 20 seconds, the train has a velocity of 0 m/s:
$ext{Area}_3 = 10 \times 0 = 0 \text{ m}$

Sum the areas:
$\text{Total Distance} = \text{Area}_1 + \text{Area}_2 + \text{Area}_3 = 48 + 16 + 0 = 64 \text{ m}$

Adding the total distance gives:
$\text{Total Distance Traveled} = 74 ext{ m}$

Step 2

Find the distance of the front of the train from the point A at the end of the 20 seconds.

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Answer

The distance of the front of the train from point A at the end of the 20 seconds can be calculated as follows:
Using the total distance calculated earlier, the distance from point A is:

$\text{Distance from A} = 74 \text{ m}$

Step 3

Find the maximum magnitude of the resultant force acting on the train.

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Answer

To find the maximum magnitude of the resultant force, we first need to calculate the maximum acceleration:

The maximum acceleration occurs during the interval from 6 seconds to 10 seconds when the train slows down from 8 m/s to 0 m/s.
The change in velocity is:
$\Delta v = 0 - 8 = -8 \text{ m/s}$
The time over which this change occurs is 4 seconds. Therefore, the maximum acceleration is:
$a_{max} = \frac{\Delta v}{\Delta t} = \frac{-8}{4} = -2 \text{ m/s}^2$
Using Newton's second law, the resultant force can be calculated as:
$F_{max} = m \cdot a_{max} = 800 \cdot -2 = -1600 \text{ N}$
Thus, the maximum magnitude of the resultant force is 1600 N.

Step 4

Explain why, in reality, the graph may not be an accurate model of the motion of the train.

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Answer

In reality, the graph does not accurately model the motion of the train for several reasons:

Abrupt Changes in Velocity: The graph shows abrupt changes in velocity, with straight lines representing instant acceleration and deceleration. In real scenarios, these changes are smoother and involve various factors such as friction and inertia.
Constant Velocity: The graph depicts a constant velocity for a prolonged duration. However, due to factors like resistance and track conditions, this is often unrealistic.
Curvature in Motion: Instead of straight lines, real motion is likely to produce curves, particularly during acceleration and deceleration phases. Thus, we would expect to see smoother transitions rather than sharp changes.

The graph below models the velocity of a small train as it moves on a straight track for 20 seconds - AQA - A-Level Maths: Pure - Question 14 - 2017 - Paper 2

Worked Solution & Example Answer:The graph below models the velocity of a small train as it moves on a straight track for 20 seconds - AQA - A-Level Maths: Pure - Question 14 - 2017 - Paper 2

Find the total distance travelled in the 20 seconds.

Find the distance of the front of the train from the point A at the end of the 20 seconds.

Find the maximum magnitude of the resultant force acting on the train.

Explain why, in reality, the graph may not be an accurate model of the motion of the train.

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