Question

A particle with positive charge \( 10^{-3} \, \text{C} \) and mass \( 0.2 \, \text{kg} \) is thrown upwards from the ground at an angle \( 45^\circ \) with the horizontal with a speed of 5 m/s. The projectile moves through a horizontal electric field of 10 V/m, which is in the same direction as the horizontal component of the initial velocity of the particle. The acceleration due to gravity is \( 10 \, \text{m/s}^2 \). The range is .............. m (Round off to three decimal places).

Hint:

When calculating the range of a projectile under the influence of both gravity and an electric field, consider both the initial velocity components and the additional force from the electric field.

Answer 1

Correct Answer: 2.51

Step 1: Initial conditions and velocity components.
The initial velocity of the projectile is \( v_0 = 5 \, \text{m/s} \), and the angle of projection is \( 45^\circ \). The horizontal and vertical components of the velocity are: \[ v_{0x} = v_0 \cos(45^\circ) = 5 \times \frac{\sqrt{2}}{2} = 3.536 \, \text{m/s}, \quad v_{0y} = v_0 \sin(45^\circ) = 5 \times \frac{\sqrt{2}}{2} = 3.536 \, \text{m/s}. \] Step 2: Electric force acting on the particle.
The horizontal electric field \( E = 10 \, \text{V/m} \) exerts a force \( F = qE \), where \( q = 10^{-3} \, \text{C} \). Thus, \[ F = 10^{-3} \times 10 = 10^{-2} \, \text{N}. \] The acceleration due to the electric field is \[ a_{\text{electric}} = \frac{F}{m} = \frac{10^{-2}}{0.2} = 5 \times 10^{-2} \, \text{m/s}^2. \] Step 3: Time of flight.
The time of flight \( t \) is given by the vertical motion equation. The total time of flight in projectile motion is \[ t = \frac{2v_{0y}}{g} = \frac{2 \times 3.536}{10} = 0.707 \, \text{s}. \] Step 4: Range.
The horizontal range \( R \) of the projectile is given by \[ R = v_{0x} \times t + \frac{1}{2} a_{\text{electric}} t^2. \] Substitute the values: \[ R = 3.536 \times 0.707 + \frac{1}{2} \times 5 \times 10^{-2} \times (0.707)^2 = 2.5 + 0.0125 = 2.5125 \, \text{m}. \] Final Answer: The range is \( 2.513 \, \text{m} \).