A person brings a mass of `1 kg` from infinity to a point `A`. Initially the mass was at rest but it moves with a speed of `2 m//s` as it reaches `A`. The work done by the person on a mass is `-3 J`. The potential of `A` is:

Options

A. -2 J/Kg
B. -4 J/kg
C. -5 J/kg
D. None of the above

Answer

To solve the problem, we need to determine the gravitational potential at point A based on the information given. Here's a step-by-step solution:

### Step 1: Understand the given data
- Mass (m) = 1 kg
- Final speed (v) = 2 m/s
- Work done by the person (W) = -3 J

### Step 2: Calculate the kinetic energy (KE) at point A
The kinetic energy (KE) of the mass when it reaches point A can be calculated using the formula:
\[ KE = \frac{1}{2} m v^2 \]
Substituting the values:
\[ KE = \frac{1}{2} \times 1 \, \text{kg} \times (2 \, \text{m/s})^2 = \frac{1}{2} \times 1 \times 4 = 2 \, \text{J} \]

### Step 3: Relate work done to potential energy and kinetic energy
According to the work-energy principle, the work done on the mass is equal to the change in potential energy (PE) plus the kinetic energy (KE):
\[ W = PE + KE \]
Rearranging this gives us:
\[ PE = W - KE \]

### Step 4: Substitute the values into the potential energy equation
Now we can substitute the values of work done and kinetic energy into the equation:
\[ PE = -3 \, \text{J} - 2 \, \text{J} = -5 \, \text{J} \]

### Step 5: Calculate the gravitational potential (V) at point A
The gravitational potential (V) at point A is defined as the potential energy per unit mass:
\[ V = \frac{PE}{m} \]
Substituting the values:
\[ V = \frac{-5 \, \text{J}}{1 \, \text{kg}} = -5 \, \text{J/kg} \]

### Final Answer
The potential at point A is \(-5 \, \text{J/kg}\).

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