Arjun has a mass of 50 kg, he eats banana of 1000 calories. If this energy is used up to lift him from the ground, the height to which he can climb is _______. Take g = 10 m/s2 and 1 calories = 4.2 Joule.
8.4 m
Calculating Height Climbed Using Energy from Food
This problem involves converting chemical energy obtained from food into gravitational potential energy to lift a mass against gravity. We are given the mass of Arjun, the energy consumed from the banana, the value of gravitational acceleration, and a conversion factor between calories and Joules.
Understanding the Concepts
- Energy Conversion: The energy from the banana is assumed to be completely converted into the energy required to lift Arjun to a certain height.
- Gravitational Potential Energy (PE): This is the energy an object possesses due to its position in a gravitational field. It is calculated using the formula: \(PE = mgh\), where \(m\) is mass, \(g\) is acceleration due to gravity, and \(h\) is height.
- Units: Energy is typically measured in Joules (J). Food energy is often measured in calories (cal) or kilocalories (kcal, often just referred to as calories on food labels). We need to convert calories to Joules using the given conversion factor.
Given Information:
- Arjun's mass (\(m\)): 50 kg
- Energy from banana: 1000 calories
- Acceleration due to gravity (\(g\)): 10 m/s\(^2\)
- Conversion factor: 1 calorie = 4.2 Joules
Step-by-Step Calculation
Step 1: Convert energy from calories to Joules.
The energy consumed is 1000 calories. Using the conversion factor, we get the energy in Joules:
Energy in Joules = Energy in calories \(\times\) Conversion factor
Energy in Joules = \(1000 \text{ calories} \times 4.2 \frac{\text{Joules}}{\text{calorie}}\)
Energy in Joules = \(4200 \text{ J}\)
Step 2: Relate energy to gravitational potential energy.
The energy from the banana is used to increase Arjun's gravitational potential energy by lifting him to a height \(h\). So, we can set the energy from the banana equal to the potential energy gained:
Energy from banana = Gravitational Potential Energy
\(4200 \text{ J} = mgh\)
Step 3: Substitute the given values into the potential energy formula.
We have \(m = 50\) kg, \(g = 10\) m/s\(^2\), and \(PE = 4200\) J. We need to find \(h\).
\(4200 \text{ J} = (50 \text{ kg}) \times (10 \text{ m/s}^2) \times h\)
\(4200 = 500 \times h\)
Step 4: Solve for the height \(h\).
Divide both sides of the equation by 500 to find \(h\):
\(h = \frac{4200}{500}\)
\(h = \frac{42}{5}\)
\(h = 8.4 \text{ m}\)
Therefore, Arjun can climb to a height of 8.4 meters using the energy from the banana.
Comparing this result with the given options, 8.4 m is one of the choices.
Revision Table: Energy and Potential Energy
| Concept | Description | Formula | Units |
|---|---|---|---|
| Energy from Food | Chemical energy available from consuming food. Often measured in calories or Joules. | Conversion: 1 calorie \(\approx\) 4.2 Joules | Calories (cal), Kilocalories (kcal), Joules (J) |
| Gravitational Potential Energy | Energy stored by an object due to its position in a gravitational field. | \(PE = mgh\) | Joules (J) |
| Mass (\(m\)) | A measure of the amount of matter in an object. | - | Kilograms (kg) |
| Acceleration due to Gravity (\(g\)) | The acceleration experienced by objects due to gravity. Varies slightly with location, often approximated as 9.8 m/s\(^2\) or 10 m/s\(^2\). | - | Meters per second squared (m/s\(^2\)) |
| Height (\(h\)) | The vertical distance of an object from a reference point. | - | Meters (m) |
Additional Information: Energy Efficiency
In reality, the conversion of energy from food to mechanical work (like climbing) is not 100% efficient. The human body is a complex system, and a significant portion of the energy from food is used for metabolic processes, maintaining body temperature, and other functions. Only a fraction of the total energy is available for physical work. This problem makes a simplifying assumption that all the energy is used for lifting, which is an ideal scenario for calculation purposes in physics problems.
The nutritional calorie (Cal or kcal) often seen on food labels is actually 1000 physics calories (cal). The problem statement uses "1000 calories", but given the result and typical physics contexts, it likely refers to 1000 physics calories, where 1 calorie = 4.2 J. If it referred to 1000 nutritional Calories (kcal), the energy would be much higher (\(1000 \times 1000 \times 4.2 = 4,200,000\) J), leading to an unrealistically high climb height.
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