A person brings a mass of `1 kg` from infinty to a point `A`. Initially, the mass was at rest but it moves at a speed of `3 m//s` as it reaches `A`. The work done by the person on the mass is `- 5.5 J`. The gravitational potential at `A` is a) -1 J/kg b) -4.5 J/kg c) -5.5 J/kg d) -10 J/kg

Options

A. `- 1 J//kg`
B. `- 4.5 J//kg`
C. `- 5.5 J//kg`
D. `- 10 J//kg`

Explanation

To find the gravitational potential at point A, we can follow these steps:

### Step 1: Calculate the initial and final energy of the mass
Initially, the mass is at rest, so its initial kinetic energy (KE_initial) is:
\[ KE_{\text{initial}} = 0 \, \text{J} \]

When the mass reaches point A, it has a speed of \(3 \, \text{m/s}\). The final kinetic energy (KE_final) can be calculated using the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \(m = 1 \, \text{kg}\) and \(v = 3 \, \text{m/s}\).
\[ KE_{\text{final}} = \frac{1}{2} \times 1 \times (3)^2 = \frac{1}{2} \times 1 \times 9 = 4.5 \, \text{J} \]

### Step 2: Use the work-energy principle
The work done on the mass by the person is given as \(-5.5 \, \text{J}\). According to the work-energy principle, the work done on the mass is equal to the change in kinetic energy plus the change in potential energy.

We can express this as:
\[ W = KE_{\text{final}} - KE_{\text{initial}} + PE_{\text{final}} - PE_{\text{initial}} \]
Since the mass is brought from infinity, we can assume the potential energy at infinity is \(0\) (PE_initial = 0). Thus, we can simplify the equation:
\[ W = KE_{\text{final}} + PE_{\text{final}} \]
Substituting the known values:
\[ -5.5 \, \text{J} = 4.5 \, \text{J} + PE_{\text{final}} \]

### Step 3: Solve for the potential energy at point A
Rearranging the equation to solve for \(PE_{\text{final}}\):
\[ PE_{\text{final}} = -5.5 \, \text{J} - 4.5 \, \text{J} = -10 \, \text{J} \]

### Step 4: Calculate the gravitational potential at point A
The gravitational potential \(V\) at point A is defined as the potential energy per unit mass:
\[ V = \frac{PE_{\text{final}}}{m} \]
Substituting the values we have:
\[ V = \frac{-10 \, \text{J}}{1 \, \text{kg}} = -10 \, \text{J/kg} \]

### Final Answer
The gravitational potential at point A is:
\[ \text{Option (d) } -10 \, \text{J/kg} \]

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