At the repair shop, the distance, \(d\) metres, a bike coasts after its brakes are released varies directly with the square of the speed, \(s\) mph, it had ...
Question 1 Report
At the repair shop, the distance, \(d\) metres, a bike coasts after its brakes are released varies directly with the square of the speed, \(s\) mph, it had when the brakes were applied. Coasting from 12 mph covers 7.2 m.
- Find the constant of proportionality. (2)
- Find the coasting distance from a speed of 20 mph. (2)
- The workshop test bay is 18 m long. State whether a bike released at 20 mph would coast clear of the bay, with a reason. (2)
Answer Details
Direct proportion to the square of the speed means coasting distance equals a constant, \(k\), multiplied by speed squared; \(k\) is found from the known test before predicting the distance at a different speed and comparing it with the length of the bay.
- \[ d = ks^2 \], and substituting \(s=12\), \(d=7.2\): \[ k = 7.2\div12^2 = 7.2\div144 = 0.05 \] [2 marks]
- With \(s=20\): \[ d = 0.05\times20^2 = 0.05\times400 = 20 \text{ m} \] [2 marks]
- Since \(20 \gt 18\), the bike released at \(20\) mph would not coast clear of the bay; it would still be moving when it reaches the far end. [2 marks]
The speed increased from \(12\) mph to \(20\) mph, a factor of \(\frac{5}{3}\), but because coasting distance depends on the square of speed, the distance increased by a factor of \(\left(\frac{5}{3}\right)^2\approx2.78\) (from \(7.2\) m to \(20\) m); this faster-than-linear growth is exactly why higher release speeds are disproportionately harder to stop safely within a fixed space.
Download The App On Google Playstore
Everything you need to excel in your exams
