A bicycle and its rider together has a mass of 62 kg. What power output of the rider is required to maintain a constant speed of 4.6 m/s (about 10.3 mph) up a 5.0% grade (a road that rises 5.0 m for every 100 m along the pavement)? Assume that frictional losses of energy are negligible.
To find the power output required by the rider to maintain a constant speed up a grade, we first need to calculate the total work done against gravity and then divide it by the time it takes to travel a given distance.
Let's break down the problem step by step:
Step 1: Calculate the weight of the bicycle and the rider.
The weight can be calculated using the formula:
Weight = mass × gravity
where the mass is given as 62 kg and the acceleration due to gravity is approximately 9.8 m/s^2.
Weight = 62 kg × 9.8 m/s^2
Weight = 607.6 N
Step 2: Calculate the work done against gravity.
The work done against gravity can be calculated using the formula:
Work = Force × Distance
In this case, the force is the weight (607.6 N) and the distance is given as a 5.0% grade, which means that for every 100 m distance, the road rises 5 m.
Distance = 100 m
Height = 5 m
The work done against gravity can be calculated as:
Work = Weight × Height
Work = 607.6 N × 5 m
Work = 3038 J
Step 3: Calculate the power output.
Power is defined as the rate at which work is done. Mathematically, it can be calculated as:
Power = Work / Time
Given that the speed is constant at 4.6 m/s, we can calculate the time it takes to travel a distance of 100 m (since the grade is given for every 100 m).
Time = Distance / Speed
Time = 100 m / 4.6 m/s
Time ≈ 21.74 s
Now, we can calculate the power output required:
Power = Work / Time
Power = 3038 J / 21.74 s
Power ≈ 139.9 W
Therefore, the rider would need to exert a power output of approximately 139.9 watts to maintain a constant speed of 4.6 m/s up a 5.0% grade.
To calculate the power output required by the rider, we can use the equation:
Power = force * velocity
First, let's determine the force required to maintain a constant speed uphill.
The force required can be divided into two components: the force due to gravity and the force due to the incline.
1. Force due to gravity:
The force due to gravity can be found using the equation:
Force due to gravity = mass * gravitational acceleration
where the mass is the combined mass of the bicycle and the rider, and the gravitational acceleration is approximately 9.8 m/s².
Force due to gravity = 62 kg * 9.8 m/s² = 607.6 N
2. Force due to the incline:
The force due to the incline can be found using the equation:
Force due to incline = mass * gravitational acceleration * sin(angle)
The angle is given as 5.0% or 0.05. So:
Force due to incline = 62 kg * 9.8 m/s² * sin(0.05) = 30.5 N
Now, let's calculate the total force required to maintain a constant speed:
Total force required = Force due to gravity + Force due to incline = 607.6 N + 30.5 N = 638.1 N
Finally, we can calculate the power output required by the rider:
Power = force * velocity = 638.1 N * 4.6 m/s = 2935.26 watts (or joules per second)
Therefore, the power output required to maintain a constant speed of 4.6 m/s uphill is approximately 2935.26 watts (or 2.94 kW).