The Question is from neet-physics electrostatic-charges-and-fields , and it was created by the author VENKATESWARARAO

Question

Two identical charged particles are projected into a uniform electric field. Particle A is projected at an angle $	heta$ with the field, while particle B is projected perpendicular to the field. If both have the same initial kinetic energy, which particle will have a greater range along the direction of the field?

VENKATESWARARAO · Accepted Answer

## CORE INFORMATION
✓ **Answer**: A
## CONCEPT FRAMEWORK
	**Primary Concept**: Projectile Motion in a Uniform Electric Field
	**Related Topics**: Kinematics, Electric Fields, Work-Energy Theorem
	**Prerequisites**: Understanding of projectile motion, electric force, and kinetic energy.
## SOLUTION PATHWAY
**Given Information**:
	- Two identical charged particles.
	- Uniform electric field.
	- Particle A projected at angle $	heta$ with the field.
	- Particle B projected perpendicular to the field.
	- Equal initial kinetic energy.
**Method & Steps**:
	1. **Resolve Initial Velocity**: Resolve particle A's initial velocity into components parallel ($v_{\parallel} = v\cos{	heta}$) and perpendicular ($v_{\perp} = v\sin{	heta}$) to the field.
		Reasoning:  Only the parallel component contributes to the range along the field direction.
	2. **Analyze Particle B**: Particle B's velocity is entirely perpendicular to the field initially. 
		Working: Its initial velocity component along the field is zero.
		Key Point: The electric field only affects the motion in its direction.
	3. **Compare Ranges**:  Particle A has a non-zero initial velocity component along the field, while particle B has zero. 
		Result: Particle A will travel further along the field direction.
		Verification: The range along the field is directly related to the initial velocity component in that direction.
## OPTION ANALYSIS
[A] ✓: Particle A has a component of its initial velocity along the field, leading to a greater range in that direction.
[B] :x: Particle B's initial velocity is entirely perpendicular to the field, resulting in zero range along the field initially. While the electric field will accelerate it along its direction, the range will be less than A due to zero initial velocity along the field.
[C] :x:  The initial velocity components along the field direction are different, hence the ranges cannot be the same.
[D] :x: While the magnitude of the electric field affects the acceleration, it doesn't change the fact that particle A has a non-zero initial velocity component along the field, unlike particle B.
## LEARNING AIDS
	**Common Mistakes**:
		1. Ignoring the vector nature of velocity:  Not resolving the velocity into components can lead to incorrect conclusions.
			How to avoid: Always consider the components of velocity in the direction of interest.
		2. Confusing range with displacement: Remember range is the distance covered in the horizontal (in this case, along the field) direction. 
			How to avoid: Clearly define the direction for which the range is being calculated.
	**Quick Tips**:
		1. For projectile motion, analyze the motion along each axis independently.
		2. The range is maximized when the launch angle is 45 degrees in the absence of other forces. In the presence of an electric field, this is no longer true.

VENKATESWARARAO · Answer

The correct Option is "Particle A"

VENKATESWARARAO · Answer

Particle B

VENKATESWARARAO · Answer

Both particles will have the same range

VENKATESWARARAO · Answer

The range depends on the magnitude of the electric field

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