A particle of mass and charge moves along the -axis in a region where the electric…

PHYSICS · Multi-part · A Level · Physics · Fields and their consequences · Fields and their consequences

Problem

A particle of mass

m

and charge

+q

moves along the

x

-axis in a region where the electric potential is given by the quadratic function

V(x) = ax^{2} + bx,

where

a = 15\,\text{V}\,\text{m}^{-2}

and

b = -12\,\text{V}\,\text{m}^{-1}

. No other forces act on the particle. At

x = 0

, the particle is projected in the

+x

direction with speed

v_0

. (a) Derive an expression for the particle’s speed

v(x)

as a function of position

x

in terms of the given constants and

v_0

. Clearly state any principles you use in your derivation. (b) Determine the position

x_{\text{max}}

at which the particle momentarily comes to rest, expressed in terms of

a

b

q

m

, and

v_0

. State the condition under which such a turning point exists. (c) Suppose

v_0 = 2.0\times10^{5}\,\text{m\,s}^{-1}

q = 1.60\times10^{-19}\,\text{C}

, and

m = 3.3\times10^{-27}\,\text{kg}

. Calculate

x_{\text{max}}

numerically and state whether the particle remains within the region

x \geq 0

. (d) Identify all position(s)

x

where the particle’s speed equals one-half of its initial speed

v_0

. Provide complete algebraic work leading to your answer and comment on the physical significance of multiple solutions, if any.