A particle of mass m and charge q is released from rest in a uniform electric field. If there is no other force on the particle, the dependence of its speed v on the distance x travelled by it is correctly given by (graphs are schematic and not drawn to scale)

Solution

Physics sir · 4 steps

Uniform field means constant acceleration, so use v² = 2ax and read v ∝ √x off it.

1. Constant force. In a uniform field the force qE is the same everywhere. With no other force acting, the acceleration a = qE/m stays constant.
2. Speed versus distance. The particle starts from rest, so the constant-acceleration equation v² = u² + 2ax has u = 0. The work-energy theorem gives the same result: qEx = 1/2mv².

   v = √(2qE/m x) ∝ √x

3. Shape of the graph. The slope is dv/dx = a/v. It is very large near x = 0, where v is small, and it keeps falling as v grows. So the curve rises steeply from the origin, then bends over and keeps rising more slowly. It never levels off.
4. So. The graph that starts steep at the origin and is concave downward is the √x curve in option (b).

The idea to keep. For constant acceleration from rest, v ∝ t but v ∝ √x. Always check whether the horizontal axis is time or distance before you pick a shape.

Answer from the NTA final answer key. Solution by PrepWiser.