Question

Three identical pucks on a horizontal air table have repelling magnets. They are held together and then released simultaneously. Each has the same speed at any instant. One puck moves due west. What is the direction of the velocity of each of the other two pucks?

Short answer

The other two pucks move 120° apart and symmetrically relative to the west direction.

Step-by-step solution

Analyze the System Symmetry

Since the three pucks are identical and they repel each other with the same force, the symmetry of the system must be considered. If one puck moves west, the other two must move in directions that keep the system balanced.

Apply Conservation of Momentum

The pucks are initially at rest, meaning the total momentum was zero. After release, the momentum of the system should also be zero. If one puck moves west with velocity \( v \), the other two must move in such a way that their momentum components cancel out the westward momentum.

Determine the Direction of Other Pucks

The pucks should move apart symmetrically. If we assume one puck moves due west, the other two must move in directions that are symmetric relative to the west direction. This means they move at 120 degrees apart from each other and the puck moving west.

Calculate the Angles

For the velocities to remain balanced, the angle between each of the two other pucks' velocity vectors and the westward direction (west-moving puck) must be 120°. This ensures that their eastward components cancel out the westward component of the first puck.

Key concepts

Symmetry in Physics

Symmetry plays an integral role in understanding the physical properties of systems, especially in situations like this one with the three pucks. The concept of symmetry in physics refers to properties that remain unchanged even when transformations, such as rotations or reflections, are applied. In our scenario with the pucks, symmetry tells us that since all pucks are identical and act on each other with the same magnitude of force once released, they will spread out uniformly from where they were initially held together.

Understanding symmetry helps us predict the behavior of the pucks without requiring complex calculations. When one puck moves west, the symmetry of the forces acting ensures that the other two pucks must move in directions that counterbalance this movement, maintaining the overall balance of the system. Essentially, the symmetry of the forces acting on each puck ensures the stability and uniform spread of motion after release, helping the system come to a new equilibrium state.

Vector Analysis

Vector analysis is crucial for solving problems involving forces and motion, like the motion of the pucks described here. Vectors carry both magnitude and direction, which is fundamental when analyzing physical scenarios. For the exercise, we treat the velocity of the puck as a vector, helping us to better understand its motion.

• The direction of the vector represents which way the puck is moving.
• The magnitude (or length) of the vector represents the speed of the puck.

To conserve momentum, if one puck moves west with a certain velocity vector, the remaining pucks must have velocity vectors that, when added together, cancel out the westward vector to keep the total system momentum unchanged. By using vector sum analysis, you can deduce that the pucks should move symmetrically at an angle of 120° to each other to achieve this balancing act.

Kinematics of Particles

Kinematics deals with the motion of particles. It focuses on the positions, velocities, and accelerations without taking into account the forces that cause the motion. In the problem with the three pucks, we are interested in their post-release motion. We want to know their velocities and directions immediately after they are released.

For the pucks, the conservation of momentum dictates that despite their individual motion, the collective center of mass remains stationary because they start from rest. Each puck is subject to purely kinematic considerations to ensure that they spread out symmetrically:

• They have the same speed because they are identical and experience the same force on release.
• The direction of each is determined by the conditions of symmetry and momentum conservation – moving at equal angles apart from each other.

Understanding kinematics allows us to reason how objects like the pucks in this thought experiment move after internal forces are applied and symmetry is restored in the system.