Two point charges q1 = q2 = 2μC are fixed at x1 = + 3 and x2 = – 3. A third particle of
mass 1 g and charge q3 = – 4 μC are released from rest at y = 4.0. Find the speed of
the particle as it reaches the origin.
q1= q
q2= -q
x
θ
To find the speed of the particle as it reaches the origin, we can use the principle of conservation of mechanical energy. The mechanical energy of the system is conserved, so we can equate the initial gravitational potential energy to the final kinetic energy of the particle.
1. Calculate the initial gravitational potential energy:
- The gravitational potential energy (U_g) at a height (h) above the reference point (usually assumed to be the ground) is given by U_g = mgh, where m is the mass and g is the acceleration due to gravity.
- In this case, the mass (m) of the particle is 1 g, which is equal to 0.001 kg, and the height (h) is 4.0 units.
- Hence, the initial gravitational potential energy (U_g) is U_g = 0.001 kg * 9.8 m/s^2 * 4.0 units.
2. Calculate the final kinetic energy:
- The final kinetic energy (K) is given by K = 0.5 * m * v^2, where v is the speed of the particle at the origin.
- We want to find the speed (v) of the particle at the origin.
3. Apply the principle of conservation of mechanical energy:
- According to the principle of conservation of mechanical energy, the initial gravitational potential energy (U_g) is equal to the final kinetic energy (K).
- So, we equate U_g to K: U_g = K.
4. Solve for the speed (v):
- Set the equation U_g = K and solve for v: 0.001 kg * 9.8 m/s^2 * 4.0 units = 0.5 * 0.001 kg * v^2.
- Simplify the equation and solve for v: 0.0392 J = 0.0005 kg * v^2.
- Divide both sides of the equation by 0.0005 kg to solve for v: v^2 = 78.4 J/kg.
- Take the square root of both sides to find v: v = √(78.4 J/kg).
5. Substitute the given values and calculate the speed (v):
- Substitute the known value for v into the equation: v = √(78.4 J/kg).
- Use a calculator to find the value of v.