A body of mass `1kg` is taken from infinity to a point P. When the body reaches that point, it has a speed of `2ms^-1`. The work done by the conservative force is `-5J`. Which of the following is true (assuming non-conservative and pseudo-forces to be absent).

Options

A. (a) Work done by the applied force is `+7J`
B. (b) The total energy possessed by the body at P is `+7J`
C. (c) The potential energy possessed by the body at P is `+5J`
D. (d) Work done by all forces together is equal to the change in kinetic energy.

Explanation

To solve the problem step by step, we will use the work-energy theorem and the information provided in the question.

### Step 1: Identify the Given Information
- Mass of the body, \( m = 1 \, \text{kg} \)
- Speed at point P, \( v = 2 \, \text{m/s} \)
- Work done by conservative force, \( W_c = -5 \, \text{J} \)

### Step 2: Calculate the Initial and Final Kinetic Energy
The initial kinetic energy (KE_initial) when the body is at infinity (starting point) is:
\[ \text{KE}_{\text{initial}} = 0 \, \text{J} \quad (\text{since the body starts from rest}) \]

The final kinetic energy (KE_final) at point P can be calculated using the formula:
\[ \text{KE}_{\text{final}} = \frac{1}{2} m v^2 \]
Substituting the values:
\[ \text{KE}_{\text{final}} = \frac{1}{2} \times 1 \, \text{kg} \times (2 \, \text{m/s})^2 = \frac{1}{2} \times 1 \times 4 = 2 \, \text{J} \]

### Step 3: Apply the Work-Energy Theorem
According to the work-energy theorem:
\[ W_{\text{total}} = \Delta KE = \text{KE}_{\text{final}} - \text{KE}_{\text{initial}} \]
Substituting the values:
\[ W_{\text{total}} = 2 \, \text{J} - 0 \, \text{J} = 2 \, \text{J} \]

### Step 4: Relate Work Done by Forces
The total work done by all forces is the sum of the work done by the conservative force and the work done by the applied force (\( W_a \)):
\[ W_{\text{total}} = W_a + W_c \]
Substituting the known values:
\[ 2 \, \text{J} = W_a - 5 \, \text{J} \]
Solving for \( W_a \):
\[ W_a = 2 \, \text{J} + 5 \, \text{J} = 7 \, \text{J} \]

### Step 5: Calculate Total Energy at Point P
The total mechanical energy at point P is the sum of kinetic energy and potential energy. The potential energy (PE) can be derived from the work done by the conservative force:
\[ \text{PE} = -W_c = -(-5 \, \text{J}) = 5 \, \text{J} \]
Thus, the total energy (E) at point P is:
\[ E = \text{KE}_{\text{final}} + \text{PE} = 2 \, \text{J} + 5 \, \text{J} = 7 \, \text{J} \]

### Conclusion
From the calculations:
- The work done by the applied force is \( 7 \, \text{J} \).
- The total energy at point P is \( 7 \, \text{J} \).
- The potential energy at point P is \( 5 \, \text{J} \).
- The work done by all forces together equals the change in kinetic energy.