To understand the logic behind why no net work is required to cause the particle to move from left to right at the same speed, let's break down the concept of work and its relation to the particle's velocity.
Work is defined as the transfer of energy resulting from the application of a force over a distance. Mathematically, the work done on an object is given by the equation:
Work = Force * Distance * cos(θ)
In this case, the force required to move the particle in the opposite direction is equal in magnitude but opposite in direction to the force that is slowing it down. As a result, the angle θ between the applied force and the displacement is 180 degrees, and the cos(180) = -1.
Now, let's consider the equation in the scenario where the particle is moving to the left at 30 m/s. The negative velocity indicates that the particle is encountering resistance or a force in the opposite direction that is causing it to decelerate. If no net force is applied, the particle will eventually come to a stop.
To make the particle move to the right at 30 m/s while experiencing the same magnitude of force, no additional work needs to be done. This is because the force applied will simply counteract the decelerating force, allowing the particle to maintain a constant velocity. Mathematically, this can be represented as:
Work = Force * Distance * cos(180) = Force * Distance * (-1) = -Force * Distance
Since the work done is the product of force and distance, and the distance is the same for both scenarios, if we multiply the force by -1, the work done becomes negative. Therefore, the net work required to cause the particle to move from left to right at the same speed is 0J (joules), signifying that no additional work is needed.
In summary, because the force required to move the particle in the opposite direction is equal and opposite to the force that is slowing it down, no additional net work is necessary to cause the particle to move from left to right at the same speed.