Wow, I guess nobody real does know how to do this problem?

Question

A positive charge of 5.20 mC is fixed in place. From a distance of 4.60 cm a particle of mass 5.60 g and charge +3.20 mC is fired with an initial speed of 72.0 m/s directly toward the fixed charge. How close to the fixed charge does the particle get before it comes to rest and starts traveling away?

**Hint: The particle comes to rest when it has no kinetic energy. The change in kinetic energy will be equal to the increase in electrical potential energy. Note that the particle has some potential energy at the starting point.

This is my work but it's still wrong for some reason.

q1 = 5.20 x 10^-6 C, q2 = 3.20 x 10^-6 C, r1 = 0.046 m, U = 72.0 m/s

εo = 9 x 10^9

the potential energy of the two charges when they are at distance r1 = 0.046 m apart is

U1 = (1/4πεo)(q1*q2/r1)

where (1/4πεo) = 9*10^9 Nm2/C2

the potential energy of the two charges when they are at distance r2 apart is

U2 = (1/4πεo)(q1*q2/r2)

then the work done is W = U2 - U1

but from the work energy thereom the work done is equal to the change in kinetic energy

U2 - U1 = ( 1/2)mV2 - (1/2)mu2

final velocity is V = 0 m/s then

U2 - U1 = -(1/2)mu^2

(1/4πεo)(q1*q2/r2) - (1/4πεo)(q1*q2/r1) = -(1/2)mu^2

1/r2 - 1/r1 = - (1/2)mu^2 / ((1/4πεo)(q1*q2))

1/r2 = 1/r1 - (1/2)mu^2 / ((1/4πεo)(q1*q2))

r2 = ___ m (i got 0.013 m)

Answer 1

your approach looks correct; this is an energy conservation problem and you can solve it either via work energy or conservation of energy

simply stated:

PE(before) +KE(before)=PE(end)+KE(end)

as you note, KE(end)=0, so that you solve for PE(end) which contains the expression for 1/r(end) and you are correct that PE for a pair of charges is kq1q2/r

without going through all your calculations, I do not see where you have gone wrong, but this is the correct way to solve for this

make sure you've got all the numbers in the right places and check again

when I grind through numbers I get 0.0083 m...not sure why we differ

Answer 2

Instead of taking differences in energies I just directly write the statement of energy conservation:

U1 + K1 = U2 + K2

(1/4πεo)(q1*q2/r1) + (1/2)mu^2 = (1/4πεo)(q1*q2/r2) + 0

Plugging in values,

9*10^9 * (q1*q2/r1) + (1/2)*(0.0056)*(72)^2 = 9*10^9 * (q1*q2/r2)

Quickmath solves this giving

r2 = 0.00842728 m

[Note:  you write "mC" but your values show you are using microcoulombs, not millicoulombs -- is that correct?]