A 74.0kg person is put on an exercise program by a physical therapist, the goal being to burn up 550 food calories in each daily session. Recall that human muscles are about 21.0% in converting the energy they use up into mechanical energy. The exercise program consists of a set of consecutive high jumps, each one 40.0cm into the air (which is pretty good for a human) and lasting 1.50s , on the average.

How many jumps should the person do per session?

To determine the number of jumps the person should do per session, we need to calculate the total mechanical energy required to burn 550 food calories.

Step 1: Convert food calories to joules.
1 food calorie = 4186 joules
So, 550 food calories = 550 * 4186 = 2,302,300 joules

Step 2: Determine the energy required for one jump.
The person's weight is 74.0 kg, and they are jumping 40.0 cm. The energy required for one jump can be calculated using the formula:
Energy = potential energy + kinetic energy

a) Calculate the potential energy:
Potential Energy = mass * gravity * height
= 74.0 kg * 9.8 m/s² * 0.40 m
= 288.16 joules

b) Calculate the kinetic energy:
The velocity at the top of the jump (where the person is momentarily at rest) is 0 m/s. Using the equation:
Kinetic Energy = (1/2) * mass * velocity²
= (1/2) * 74.0 kg * (0 m/s)²
= 0 joules

Therefore, the total energy required for one jump is:
Total Energy = Potential Energy + Kinetic Energy
= 288.16 joules + 0 joules
= 288.16 joules

Step 3: Account for the efficiency of human muscles.
Given that human muscles are about 21.0% efficient, we divide the total energy required for one jump by the efficiency:
Adjusted Energy = Total Energy / Efficiency
= 288.16 joules / 0.21
= 1372.19 joules

Step 4: Determine the number of jumps.
To find the number of jumps, we divide the total energy needed to burn 550 food calories by the adjusted energy required for one jump:
Number of jumps = Total Energy / Adjusted Energy
= 2,302,300 joules / 1372.19 joules
≈ 1679 jumps

Therefore, the person should do approximately 1679 jumps per session.

To determine the number of jumps the person should do per session, we need to calculate the mechanical energy generated by each jump and then divide the total energy required by the energy generated per jump.

First, let's calculate the mechanical energy generated by each jump:
- The height of each jump is 40.0 cm, which is equivalent to 0.40 m.
- The person's mass is 74.0 kg, and the acceleration due to gravity is approximately 9.8 m/s².
- The potential energy gained during each jump can be calculated using the formula: Potential Energy (PE) = mass × gravity × height.

PE = 74.0 kg × 9.8 m/s² × 0.40 m
PE = 288.96 J (joules)

Since human muscles convert only 21.0% of the energy they use into mechanical energy:
Mechanical Energy Generated (MEG) = 0.21 × Potential Energy

MEG = 0.21 × 288.96 J
MEG = 60.68 J (joules)

Now, we can calculate the number of jumps required per session:
- The total energy required per session is 550 food calories.
- To convert calories to joules, multiply calories by 4.184: Energy (E) = 550 calories × 4.184 J/cal

E = 2300.2 J (joules)

To find the number of jumps, we divide the required energy by the energy generated per jump:

Number of Jumps = Energy Required ÷ Mechanical Energy Generated

Number of Jumps = 2300.2 J ÷ 60.68 J
Number of Jumps ≈ 37.89

Therefore, the person should do approximately 38 jumps per session.