How Much Power Does a Cyclist Develop Climbing a Slope with Resistance?

[Mjpcampbell]

Homework Statement

A man of mass 70kg rides a bicycle of mass 15kg at a steady speed of 4.0 ms^-1 up a road which rises 1.0m for every 20m of its length. What power is the cyclist developing if there is a constant resistance to motion of 20 N? Take g as 10 ms^-2 if necessary.

(The answer given in the book is 2.5x10²W)

Homework Equations

power = work done/time taken
work done = force x distance moved in direction of force = force x velocity
w= mg

The Attempt at a Solution

I really can't get anywhere with this one after about 45 mins. I know I will kick myself. At first I made an attempt to find the angle of the slop using trigonometry and got a value of 2.9 degrees but this got me nowhere. I'm pretty sure I need to multiply the combined mass of 85 kg by the 1m rise in order to find the value of the work done against gravity. 85 x 10 = 850 N. I tried constructing a right-angled triangle with a hypotenuse of 20m and gave the side with a 1m rise a value of 850 N. But again this got me nowhere. I think the problem is calculating the force on the flat when the cyclist is traveling at 4ms^-1 but I don't know how to do this in order to add that value onto the vertical component. My humble, frazzled brain could use some assistance.

 

[jegues]

I'm not sure if this is correct but I'll throw it out there, maybe someone else can help justify or disprove this claim.

I think the right angle triangle approach would work. You know he's traveling at a constant speed and you know the height he rises over a given distance.

All the work as he travels across the hypotenuse of this triangle will be a summation of kinetic energy and potential energy. Now solve for the time it takes the rider to travel up the hypotenuse of the triangle and hopefully you'll generate the correct answer.

Again I haven't done any of the calculations so I could be wrong, just thought I'd throw it out there.

 

[kerimek]

I would approach this problem by first identifying the known values and variables, and then using the appropriate equations to solve for the unknown variable (power).

Known values:
- Mass of man (m1) = 70 kg
- Mass of bicycle (m2) = 15 kg
- Speed (v) = 4.0 ms-1
- Slope of road (rise/run) = 1/20
- Constant resistance to motion (Fr) = 20 N
- Acceleration due to gravity (g) = 10 ms-2

Unknown variable:
- Power (P)

First, I would calculate the total mass of the system (m1+m2) = 85 kg.

Next, I would calculate the work done against gravity (Wg) using the equation W = mgh, where h is the vertical distance moved (1m) and g is the acceleration due to gravity. Therefore, Wg = 85 kg x 10 ms-2 x 1m = 850 J.

Then, I would calculate the work done against resistance (Wr) using the equation W = Fd, where F is the force of resistance (20 N) and d is the distance moved (20m). Therefore, Wr = 20 N x 20m = 400 J.

Next, I would calculate the total work done (W) by adding the work done against gravity and resistance: W = Wg + Wr = 850 J + 400 J = 1250 J.

Using the equation for power, P = W/t, I would then solve for time (t) by rearranging the equation t = W/P. Plugging in the values for W and P, I get t = 1250 J/2.5x102 W = 0.005 s.

Finally, I would plug in the values for work (W) and time (t) into the equation for work done (W = Fd) to solve for the force (F) acting on the cyclist on the flat road. Therefore, F = W/d = 1250 J/20m = 62.5 N.

To find the total force (Ft) acting on the cyclist, I would use the Pythagorean theorem to calculate the hypotenuse of the right triangle formed by the forces of gravity (850 N) and the force on the flat road (62.5 N