A bullet of mass `2 gm` is having a charge of `2 muc`. Through what potential difference must it be accelerated, starting from rest, to acquire a speed of `10 m//s`
(a) `5kV` (b) `50 kV` (c) `5V` (d) `50V`

Options

A. `5kV`
B. `50 kV`
C. `5V`
D. `50V`

Answer

To solve the problem, we need to find the potential difference required to accelerate a bullet of mass 2 grams and charge 2 microcoulombs to a speed of 10 m/s. We will use the relationship between kinetic energy and electric potential energy.

### Step-by-Step Solution:

1. **Convert the mass from grams to kilograms:**
\[ \text{Mass} (m) = 2 \, \text{grams} = 2 \times 10^{-3} \, \text{kg} \]

**Hint:** Remember that 1 gram = 0.001 kg.

2. **Convert the charge from microcoulombs to coulombs:**
\[ \text{Charge} (Q) = 2 \, \mu C = 2 \times 10^{-6} \, C \]

**Hint:** Recall that 1 microcoulomb (μC) = \(10^{-6}\) coulombs.

3. **Use the formula for kinetic energy (KE):**
The kinetic energy of the bullet when it reaches a speed \(v\) is given by:
\[ KE = \frac{1}{2} m v^2 \]
Substituting the values:
\[ KE = \frac{1}{2} \times (2 \times 10^{-3}) \times (10)^2 \]
\[ KE = \frac{1}{2} \times (2 \times 10^{-3}) \times 100 = 0.1 \, \text{J} \]

**Hint:** Remember to square the velocity and multiply by half the mass.

4. **Relate kinetic energy to electric potential energy:**
The change in electric potential energy (PE) when the charge is accelerated through a potential difference \(V\) is given by:
\[ PE = QV \]
Setting the change in kinetic energy equal to the change in potential energy:
\[ QV = KE \]
Therefore, we can express the potential difference \(V\) as:
\[ V = \frac{KE}{Q} \]

**Hint:** This step uses the principle of conservation of energy.

5. **Substituting the values to find the potential difference:**
\[ V = \frac{0.1 \, \text{J}}{2 \times 10^{-6} \, C} \]
\[ V = \frac{0.1}{2 \times 10^{-6}} = 5 \times 10^{4} \, V = 50 \, kV \]

**Hint:** When dividing, ensure to handle the powers of ten correctly.

6. **Final Answer:**
The potential difference required is:
\[ V = 50 \, kV \]

Thus, the correct option is (b) \(50 kV\).