need help with this problem
A charge of -2.73 uC is fixed in place. From a horizontal distance of 0.0465 m, a particle of mass 9.50 x 10-3 kg and charge -9.56 uC is fired with an initial speed of 97.2 m/s directly toward the fixed charge. What is the distance of closest approach?
To find the distance of closest approach, you can use the principle of conservation of mechanical energy and the principle of conservation of electric potential energy.
1. Calculate the electric potential energy at the initial position:
The electric potential energy between two charges is given by the equation:
PE = (k * q1 * q2) / r
Where:
- PE is the electric potential energy
- k is the electrostatic constant (k = 9 * 10^9 N*m^2/C^2)
- q1 and q2 are the charges of the two particles
- r is the distance between the two particles
Substituting the given values, we get:
PE_initial = (9 * 10^9 N*m^2/C^2) * (-2.73 * 10^-6 C) * (-9.56 * 10^-6 C) / 0.0465 m
2. Calculate the initial kinetic energy:
Since the particle is being fired with an initial speed of 97.2 m/s, the initial kinetic energy is given by:
KE_initial = (1/2) * m * v^2
Substituting the given values, we get:
KE_initial = (1/2) * (9.50 * 10^-3 kg) * (97.2 m/s)^2
3. At the distance of closest approach, the electric potential energy will be converted into kinetic energy. Thus, we equate the initial electric potential energy to the final kinetic energy:
PE_initial = KE_final
4. Calculate the distance of closest approach:
Since the particle is moving directly toward the fixed charge, the final velocity at the distance of closest approach is in the opposite direction of the initial velocity. Therefore, the final kinetic energy is given by:
KE_final = (1/2) * m * (-v)^2
Substituting the given values, we get:
KE_final = (1/2) * (9.50 * 10^-3 kg) * (-97.2 m/s)^2
Now, set the initial potential energy equal to the final kinetic energy and solve for the distance of closest approach:
(9 * 10^9 N*m^2/C^2) * (-2.73 * 10^-6 C) * (-9.56 * 10^-6 C) / 0.0465 m = (1/2) * (9.50 * 10^-3 kg) * (-97.2 m/s)^2
Solve this equation to find the distance of closest approach.