Two 2.0-mm-diameter beads, C and D, are 10 mm apart, measured between their centers. Bead C has mass 1.0 g and charge 2.0 nC. Bead D has mass 2.0 g and charge −1.0 nC . If the beads are released from rest, what are the speeds vc and vD at the instant the beads collide?

Verified step by step guidance

1. Identify the charges and masses of the beads. Bead C has a charge of +2.0 nC and a mass of 1.0 g, while Bead D has a charge of -1.0 nC and a mass of 2.0 g.
2. Calculate the initial electrostatic force between the beads using Coulomb's Law: $F=k\frac{|q_1{q}_2|}{r^2}$, where $k$ is Coulomb's constant, $q_1$ and $q_2$ are the charges on the beads, and $r$ is the initial separation between the centers of the beads.
3. Apply Newton's second law to each bead to find their accelerations. For Bead C: $a_C=\frac{F}{m_C}$ and for Bead D: $a_D=\frac{F}{m_D}$, where $F$ is the magnitude of the electrostatic force, and $m_C$ and $m_D$ are the masses of beads C and D, respectively.
4. Since the beads start from rest and move towards each other, use the kinematic equation $v^2=2a\mathrm{\Delta }x$ to find the speeds of the beads at the instant they collide, where $v$ is the final speed, $a$ is the acceleration, and $\mathrm{\Delta }x$ is the displacement (half the initial separation since they meet in the middle).
5. Finally, calculate the individual speeds of each bead at the moment of collision using the accelerations found in step 3 and the displacement from step 4.

Key Concepts

Coulomb's Law

Coulomb's Law describes the electrostatic force between two charged objects. It states that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between their centers. This principle is crucial for calculating the initial force acting on the beads due to their charges before they collide.

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant. In this scenario, the potential energy due to the electrostatic force between the beads will convert into kinetic energy as they move towards each other. Understanding this concept allows us to relate the initial potential energy to the final kinetic energy of the beads at the moment of collision.

Kinematics and Velocity

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. In this problem, we need to apply kinematic equations to determine the speeds of the beads at the moment of collision. The relationship between distance, time, and velocity will help us find the final speeds of beads C and D as they collide.