A bicyclist is coasting straight down a hill at a constant speed. The mass of the rider and bicycle is 85.0 kg, and the hill is inclined at 19.0° with respect to the horizontal. Air resistance opposes the motion of the cyclist. Later, the bicyclist climbs the same hill at the same constant speed. How much force (directed parallel to the hill) must be applied to the bicycle in order for the bicyclist to climb the hill?

A bicyclist of mass 66 (including the bicycle) can coast down a 5.0 hill at a steady speed of because of air resistance.How much force must be applied to climb the hill at the same speed and same air resistance?

To determine the force required to climb the hill, we need to analyze the forces acting on the bicyclist.

1. Weight force: This force acts vertically downwards and can be calculated using the formula: weight = mass * gravity. In this case, the mass is given as 85.0 kg, and the acceleration due to gravity is approximately 9.8 m/s^2. Therefore, weight = 85.0 kg * 9.8 m/s^2 = 833 N.

2. Normal force: This force acts perpendicular to the surface of the hill. Since the bicycle is on an inclined plane, the normal force can be calculated using the formula: normal force = weight * cos(angle of inclination). In this case, the angle of inclination is given as 19.0°. Therefore, normal force = 833 N * cos(19.0°) = 788 N.

3. Friction force: This force acts parallel to the surface of the hill and opposes the motion of the bicyclist. The friction force can be calculated using the formula: friction force = coefficient of friction * normal force. However, since the question states that the bicyclist is coasting down the hill at a constant speed, we know that the net force acting on the cyclist must be zero and therefore, the friction force must be equal in magnitude but opposite in direction to the force applied by the air resistance.

4. Force applied by air resistance: This force acts opposite to the motion of the cyclist and has the same magnitude as the friction force. Therefore, the force applied by air resistance is also 788 N.

When the bicyclist climbs the hill at the same constant speed, the force applied by air resistance must be overcome by an external force.

5. Force required to climb the hill: Since the bicyclist is moving at a constant speed, the net force in the horizontal direction must be zero. This means that the force required to climb the hill must be equal in magnitude but opposite in direction to the force of air resistance. Therefore, the force required to climb the hill is also 788 N.

So, in order for the bicyclist to climb the hill at the same constant speed, a force of 788 N (directed parallel to the hill) must be applied to the bicycle.

To determine the force required to climb the hill, we need to consider the forces acting on the bicyclist when coasting downhill and when climbing uphill.

When coasting downhill at a constant speed, the only force acting on the bicyclist in the horizontal direction (parallel to the hill) is the force of air resistance since there is no applied force. According to Newton's second law, the net force acting on an object is equal to its mass multiplied by its acceleration. Since the speed is constant, the acceleration is zero, which means the net force is also zero. Therefore, the force of air resistance must be equal in magnitude and opposite in direction to the force of gravity acting on the bicyclist, causing them to cancel out.

The force of gravity is equal to the mass of the rider and bicycle multiplied by the acceleration due to gravity (9.8 m/s²). Therefore, the force of air resistance when coasting downhill is 85.0 kg * 9.8 m/s² = 833 N.

When climbing the hill at the same constant speed, the force of air resistance remains the same. However, now we need to consider the force required to counteract the component of gravity pulling the bicyclist down the hill. This component of gravity can be determined by multiplying the force of gravity (mass * acceleration due to gravity) by the sine of the angle of inclination (19.0°).

So, the force required to climb the hill is equal to the force of air resistance plus the component of gravity pulling the bicyclist down the hill: F = F_air_resistance + (mass * acceleration due to gravity * sin(angle of inclination)).

Substituting the given values, we have: F = 833 N + (85.0 kg * 9.8 m/s² * sin(19.0°)).

Calculating this expression will give us the force required to climb the hill.