SOLUTION: Pleaseee Help me with the following question, I will be grately thankful.
A bicycle has two sets of gears, one at the pedals and one at the rear wheel. A typical 10-speed bicycl
Question 202242: Pleaseee Help me with the following question, I will be grately thankful.
A bicycle has two sets of gears, one at the pedals and one at the rear wheel. A typical 10-speed bicycle has two gears on the chain wheel and five gears on the rear wheel. The speed at which a bicycle travels depends on three independent factors:
- The first is the speed at which the cyclist pedals to turn the front gear (measured in rotations per minute).
- The second is the gear ratio from the front gear to the rear wheel (a ratio between the number of teeth on the rear gear).
- The third is the size of the rear wheel (measured as the diameter of the wheel).
a)Develop a formula that predicts the velocity in km/h for a bicyclist.(2 marks)
b)How fast will a cyclist travel (in km/h) who is pedalling at 50 rpm with a 42-toothed gear on the front, a 14-toothed gear on the back, and a tire with a diameter of 26 inches?(2 marks)
c)In what ways can a cyclist pedal more slowly but maintain the same speed?(1 mark) Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website! let x = number of gear teeth in front.
let y = number of gear teeth in back.
let d = diameter of the back wheel.
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let kmph = distance traveled in kilometers per hour.
let r = radius of the back wheel in inches = d/2
let c = circumference of the back wheel in inches.
let k = inch to kilometer ratio.
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circumference of a circle = . this is also the same as since r = d/2.
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rpm = number of revolutions of the front crank per minute.
rph = number of revolutions of the front crank per hour = 60 * rpm.
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your formula looks like it should be:
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x = 42
y = 14
d = 26
r = d/2 = 13
c =
k = .0254/1000
rpm = 50
rph = 50 * 60 = 3000
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your formula becomes the following, using these values:
solving this equation, you get an answer of:
18.6723701 which is the speed of the bicycle in kilometers per hour.
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here's how it works.
the front crank is roating at 50 revolutions per minute.
times that by 60 and it becomes 3000 revolutions per hour.
the number of teeth in the front crank is 42 and the number of teeth in the rear crank is 14. 1 revolution of the front crank causes the rear crank to revolve 3 times (42/14).
this means that 3000 revolutions per hour of the front crank results in 9000 revolutions per hour of the rear crank per hour.
each revolution of the rear crank cause the rear wheel to revolve one time.
the circumference of the rear wheel is a function of the diameter. since and = d, c = expresses the same relationship because = d. bottom line is 1 revolution of the rear wheel causes the bicycle to travel inches, or inches which is the same thing and equals 81.68...... inches.
this means that 9000 revolutions of the rear wheel per hour causes the bicycle to move 9000 * 81.68.... inches = 735132.68 ... inches.
the ratio of inches to kilometers is .0254/1000.
multiplying 735132.68 ... inches by .0254/1000 results in 18.67237 kmph (kilometers per hour).
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the answer looks reasonable.
the miles per hour would be approximately .625 * 18.67237 = 11.67 mph which is a reasonable speed to assume for a bicycle.
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if the cyclist wants to pedal more slowly and achieve the same speed, he can do this by two basic ways.
1. increase the ratio of the front gear teeth to the back gear teeth.
2. increase the diameter of the rear wheel.
either one of these will increase the distance traveled per revolution of the front crank.
