Just before your mountain bike ride, you eat a 240 Calorie exercise bar. (You can find the conversion between food Calories and Joules in the People’s Physics chapter and several references online.) The carbon bonds in the food are broken down in your stomach, releasing energy. About half of this energy is lost due to inefficiencies in your digestive system.
Given the losses in your digestive system how much of the energy, in Joules, can you use from the exercise bar? __________________________
After eating, you climb a 500 m hill on your bike. The combined mass of you and your bike is 75 kg.
How much gravitational potential energy has been gained by you and your bike? __________________________________________________________
Where did this energy come from? ______________________________________
If you ride quickly down the mountain without braking but losing half the energy to air resistance, how fast are you going when you get to the bottom? _________________
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To find the amount of energy that can be used from the exercise bar, we need to consider the fact that about half of the energy is lost due to inefficiencies in the digestive system. Let's assume that the exercise bar provides 240 Calories and we need to convert this to Joules.
1 Calorie is equal to 4.18 Joules. Therefore, 240 Calories would be equal to 240 * 4.18 = 1003.2 Joules.
Since half of the energy is lost, we need to calculate half of the energy provided by the exercise bar. Therefore, the amount of energy that can be used is 1003.2 Joules / 2 = 501.6 Joules.
Next, let's calculate the gravitational potential energy gained while climbing the 500 m hill on your bike. The gravitational potential energy can be calculated using the formula:
Gravitational Potential Energy = mass * acceleration due to gravity * height
The mass of you and your bike is given as 75 kg, the acceleration due to gravity is approximately 9.8 m/s^2, and the height of the hill is 500 m.
Gravitational Potential Energy = 75 kg * 9.8 m/s^2 * 500 m = 367,500 Joules
So, you and your bike have gained 367,500 Joules of gravitational potential energy.
The energy for climbing the hill comes from the conversion of chemical energy in your body into mechanical energy to pedal the bike and overcome the force of gravity.
If you ride quickly down the mountain without braking but lose half of the energy to air resistance, we can calculate the final speed at the bottom of the mountain using the conservation of energy principle.
The initial energy at the top of the mountain is the gravitational potential energy gained, which is 367,500 Joules. However, we lose half of this energy due to air resistance. So, the remaining energy is 367,500 Joules / 2 = 183,750 Joules.
We can convert this energy into kinetic energy using the formula:
Kinetic Energy = (1/2) * mass * velocity^2
The mass of you and your bike is 75 kg.
183,750 Joules = (1/2) * 75 kg * velocity^2
Simplifying the equation:
velocity^2 = (2 * 183,750 Joules) / 75 kg
velocity^2 = 2,450 Joules / kg
Taking the square root of both sides:
velocity = √(2,450 Joules / kg) = 19.7 m/s
Therefore, when you reach the bottom of the mountain, without braking and losing half the energy to air resistance, you would be traveling at a speed of approximately 19.7 m/s.