To solve the problem step by step, we will analyze the motion of the particle under the influence of both gravity and the electric field.
### Step 1: Identify the given values
- Mass of the particle, \( m = 2 \, \text{kg} \)
- Charge of the particle, \( q = 1 \, \text{mC} = 1 \times 10^{-3} \, \text{C} \)
- Initial vertical velocity, \( u_y = 10 \, \text{m/s} \)
- Electric field strength, \( E = 10^4 \, \text{N/C} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \) (assuming standard value)
### Step 2: Calculate the time of flight
The total time of flight for a projectile launched vertically can be calculated using the formula:
\[
t = \frac{2u_y}{g}
\]
Substituting the values:
\[
t = \frac{2 \times 10}{10} = 2 \, \text{s}
\]
### Step 3: Calculate the horizontal acceleration
The force acting on the charged particle due to the electric field is given by:
\[
F = qE
\]
The acceleration \( a_x \) in the horizontal direction can be calculated using Newton's second law \( F = ma \):
\[
a_x = \frac{F}{m} = \frac{qE}{m}
\]
Substituting the values:
\[
a_x = \frac{(1 \times 10^{-3}) \times (10^4)}{2} = \frac{10}{2} = 5 \, \text{m/s}^2
\]
### Step 4: Calculate the horizontal range
The horizontal range \( R \) can be calculated using the formula:
\[
R = u_x t + \frac{1}{2} a_x t^2
\]
Since the initial horizontal velocity \( u_x = 0 \):
\[
R = 0 + \frac{1}{2} a_x t^2
\]
Substituting the values:
\[
R = \frac{1}{2} \times 5 \times (2^2) = \frac{1}{2} \times 5 \times 4 = 10 \, \text{m}
\]
### Step 5: Calculate the maximum height
The maximum height \( H \) can be calculated using the formula:
\[
H = \frac{u_y^2}{2g}
\]
Substituting the values:
\[
H = \frac{10^2}{2 \times 10} = \frac{100}{20} = 5 \, \text{m}
\]
### Final Results
- Total time of flight: \( t = 2 \, \text{s} \)
- Horizontal range: \( R = 10 \, \text{m} \)
- Maximum height: \( H = 5 \, \text{m} \)
### Summary of Answers
- The total time of flight is \( 2 \, \text{s} \).
- The horizontal range is \( 10 \, \text{m} \).
- The maximum height is \( 5 \, \text{m} \).
