A positive charge of 4.10 C is fixed in place. From a distance of 3.70 cm a particle of mass 6.80 g and charge +3.60 C is fired with an initial speed of 80.0 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?
To determine how close the particle gets to the fixed charge before coming to rest and starting to travel away, we can calculate the distance using the principle of conservation of mechanical energy.
The initial kinetic energy of the particle is equal to the potential energy when it comes to rest and starts traveling away. Therefore, we can equate the initial kinetic energy to the final potential energy.
1. Calculate the initial kinetic energy of the particle:
KE_initial = (1/2) * mass * (initial velocity)^2
KE_initial = (1/2) * 0.00680 kg * (80.0 m/s)^2
2. Calculate the final potential energy of the particle:
PE_final = (k * q1 * q2) / r
Here, k is the Coulomb's constant (8.99 x 10^9 N m^2/C^2),
q1 is the charge of the fixed charge (4.10 C),
q2 is the charge of the particle (+3.60 C), and
r is the distance between the charges.
Since the particle comes to rest and starts traveling away, the final potential energy is zero.
PE_final = 0
3. Equate the initial kinetic energy to the final potential energy:
KE_initial = PE_final
(1/2) * 0.00680 kg * (80.0 m/s)^2 = 0
Solve for the distance, r.
r = (k * q1 * q2) / (2 * KE_initial)
r = (8.99 x 10^9 N m^2/C^2) * (4.10 C) * (3.60 C) / (2 * (1/2) * 0.00680 kg * (80.0 m/s)^2)
r = (8.99 x 10^9 N m^2/C^2) * (4.10 C) * (3.60 C) / (0.00680 kg * (80.0 m/s)^2)
r = 6.818803977 m
Therefore, the particle gets as close as approximately 6.82 meters to the fixed charge before coming to rest and starts traveling away.
To find the distance at which the particle comes to rest and starts traveling away from the fixed charge, we need to consider the electrostatic force and the initial kinetic energy of the particle.
1. Calculate the electrostatic force (F) between the fixed charge and the moving particle.
- The electrostatic force between two charges can be calculated using Coulomb's law: F = k * (q1 * q2) / r^2
- Here, k is the electrostatic constant (9 * 10^9 Nm^2/C^2), q1 is the charge of the fixed charge (4.10 C), q2 is the charge of the moving particle (+3.60 C), and r is the distance between them (3.70 cm = 0.0370 m).
Plugging in the values:
F = (9 * 10^9 Nm^2/C^2) * (4.10 C) * (+3.60 C) / (0.0370 m)^2
F = 32457621.6 N.
2. Calculate the initial kinetic energy (K) of the moving particle.
- The initial kinetic energy of the particle can be calculated using the equation: K = (1/2) * m * v^2
- Here, m is the mass of the particle (6.80 g = 0.00680 kg), and v is the initial speed of the particle (80.0 m/s).
Plugging in the values:
K = (1/2) * (0.00680 kg) * (80.0 m/s)^2
K = 21.760 J.
3. When the particle comes to rest and starts traveling away, the electrostatic force acting on the particle will be equal to the opposing force due to the kinetic energy.
- At this point, the electrostatic force opposes the motion of the particle and brings it to rest.
- The force opposing the motion is given by: F = -K / d (opposite direction to the motion)
- Here, d is the distance at which the particle comes to rest.
Setting the forces equal to each other:
32457621.6 N = -21.760 J / d
-32457621.6 N * d = -21.760 J
d = -21.760 J / -32457621.6 N
d ≈ 6.7 x 10^-7 meters.
Therefore, the particle comes closest to the fixed charge at a distance of approximately 6.7 x 10^-7 meters before it comes to rest and starts traveling away.