If a bicyclist of mass 68.0 kg (including the bicycle) can coast down a 7.00° hill at a steady speed of 6.20 km/h because of air resistance, how much force must be applied to climb the hill at the same speed (and the same air resistance)?

Weight component downhill = air resistance at 6.2 km/h

(You know that from the constant coasting speed)

When going uphill at the same speed, the uphill force must equal the sum of the weight downhill component and the air resistance. That sum will be TWICE the weight component downhill.

F = 2 M g sin7 = 81.2 N

To find the force required to climb the hill at the same speed, we need to consider the gravitational force and the air resistance.

1. Gravitational force: The force due to gravity can be calculated using the formula F = m * g, where m is the mass and g is the acceleration due to gravity (9.8 m/s²). In this case, the mass of the bicyclist is 68.0 kg. Therefore, the gravitational force is F_grav = 68.0 kg * 9.8 m/s².

2. Air resistance force: The air resistance force can be calculated using the formula F_air = 0.5 * ρ * A * C_d * v², where ρ is the air density, A is the frontal area, C_d is the drag coefficient, and v is the velocity of the bicyclist. Since we want to maintain the same speed, the air resistance force will be the same as during the coasting down the hill.

To calculate the force required to climb the hill at the same speed, let's assume the air resistance force is F_air = X N.

Therefore, the total force required to climb the hill at the same speed is:

F_total = F_grav + F_air
= (68.0 kg * 9.8 m/s²) + X N

Now, let's calculate the total force required to climb the hill at the same speed.

To determine the force required to climb the hill at the same speed as when coasting down, we can use Newton's laws of motion.

First, let's find the force of gravity acting on the bicyclist while coasting down the hill. The force of gravity can be calculated using the formula:

Force of Gravity = Mass * Acceleration due to gravity

Given that the mass of the bicyclist (including the bicycle) is 68.0 kg and the acceleration due to gravity is approximately 9.8 m/s², we can calculate:

Force of Gravity = 68.0 kg * 9.8 m/s² = 666.4 N

Now, let's find the force of air resistance acting on the bicyclist while coasting down the hill. The force of air resistance can be a bit tricky to determine accurately, as it depends on various factors like the shape of the bicycle and the speed. However, for simplicity, let's assume that the force of air resistance is equal to the force of gravity while coasting down the hill.

Therefore, the force of air resistance = 666.4 N

Now, to find the total force needed to climb the hill at the same speed, we need to consider that the force of gravity is working against the objective motion. So, to determine the total force required, we need to subtract the force of air resistance from the force of gravity:

Total Force = Force of Gravity - Force of Air Resistance
Total Force = 666.4 N - 666.4 N = 0 N

Hence, the force required to climb the hill at the same speed and with the same air resistance is zero, meaning no additional force needs to be applied.