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A car moves along a horizontal straight road, passing two points A and B - Edexcel - A-Level Maths Mechanics - Question 3 - 2008 - Paper 1
Question 3
A car moves along a horizontal straight road, passing two points A and B. At A the speed of the car is 15 m s⁻¹. When the driver passes A, he sees a warning sign W a... show full transcript
Worked Solution & Example Answer:A car moves along a horizontal straight road, passing two points A and B - Edexcel - A-Level Maths Mechanics - Question 3 - 2008 - Paper 1
Step 1
Sketch the speed-time graph
Answer
To construct the speed-time graph, we start by plotting key points based on the problem statement.
- At point A, the initial speed is 15 m/s.
- From A to W, the car decelerates uniformly to 5 m/s. This portion of the graph will be a sloping line descending from 15 m/s down to 5 m/s.
- At point W, the speed remains constant at 5 m/s for a duration of 16 seconds.
- After 16 seconds, the car accelerates back to a speed of V m/s.
- After reaching V, the car moves at this constant speed for 22 seconds until it reaches point B.
Ensure the graph accurately reflects these changes in speed over time.
Step 2
Find the time taken for the car to move from A to B
Answer
To find the total time taken for the car to move from A to B, we need to calculate the time during each segment of the journey.
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From A to W: Using the formula for uniform acceleration, we need to find the time taken to decelerate from 15 m/s to 5 m/s. Assuming uniform deceleration, we can use the average speed:
- Average speed = rac{15 + 5}{2} = 10 ext{ m/s}.
- Distance from A to W = 120 m.
- Time t = rac{ ext{Distance}}{ ext{Average Speed}} = rac{120}{10} = 12 seconds.
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From W to the final speed V: This segment lasts for 16 seconds (as specified in the problem).
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From V to B: The car travels the remainder of the distance at speed V for 22 seconds.
Total time = Time from A to W + Time from W to V + Time from V to B = 12 + 16 + 22 = 50 seconds.
Step 3
The distance from A to B is 1 km
Answer
We convert the distance from kilometers to meters for calculation:
- Distance = 1 km = 1000 m.
Using the average speed during each segment:
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From A to W: Time = 12 seconds, Average speed = 10 m/s, Distance = .
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From W to V: Remaining distance after reaching W to B can be calculated. The distance covered during acceleration should equal the distance from W to the point where speed becomes V.
If we denote the total distance from W to B as , we calculate:
- So we have 120 (A to W) + D = 1000.
- D must cover the distance traveled at speed V for 22 seconds.
- From the previous time calculations, .
Thus the equation becomes: .
After rearranging: . Solving for V gives V = rac{880}{22} = 40 ext{ m/s}.
Step 4
Find the value of V
Answer
Given the analysis above, we already determined that the speed V can be derived from the distance relationship we established from W to B.
Substituting and solving the previous equation directly leads to: Solving gives:
ightarrow V = rac{880}{22} = 40 ext{ m/s} $$. Thus, the value of V is 28 m/s.