Question

If the net work done on a particle is zero, what can be said about the particle's speed?

Short answer

Answer: If the net work done on a particle is zero, its final speed will either be equal to or the negative of its initial speed. This means the particle's speed will not change or it will change direction.

Step-by-step solution

Recall the work-energy theorem

The work-energy theorem states that the net work done on an object is equal to its change in kinetic energy. Mathematically, this can be written as: $$W_{net} = \Delta KE$$ where \(W_{net}\) is the net work done on the object and \(\Delta KE\) represents the change in kinetic energy of the object.

Consider the case when the net work done is zero

The exercise asks us to analyze the situation when the net work done on a particle is zero. In that case, the work-energy theorem becomes: $$0 = \Delta KE$$

Calculate the change in kinetic energy

The kinetic energy of an object is given by the formula: $$KE = \frac{1}{2}mv^2$$ where \(m\) is the mass of the object and \(v\) is its speed. The change in kinetic energy, \(\Delta KE\), is the difference between the final and initial kinetic energies: $$\Delta KE = KE_f - KE_i = \frac{1}{2}m(v_f^2 - v_i^2)$$

Substitute the change in kinetic energy in the work-energy theorem

Plug the expression for \(\Delta KE\) into the work-energy theorem equation when the net work is zero: $$0 = \frac{1}{2}m(v_f^2 - v_i^2)$$

Determine the relationship between initial and final speeds

In order to satisfy the equation when the net work is zero, the relationship between the initial and final speeds must be such that: $$v_f^2 - v_i^2 = 0$$ Solving for \(v_f\), we get: $$v_f = \pm v_i$$ In conclusion, if the net work done on a particle is zero, its final speed will either be equal to or the negative of its initial speed. This means the particle's speed will not change (if \(v_f = v_i\)) or it will change direction (if \(v_f = -v_i\)).

Key concepts

Kinetic Energy

When discussing the motion of objects, the term kinetic energy is central. It is defined as the energy that an object possesses due to its motion. Mathematically, kinetic energy (\(KE\)) is expressed through the equation:
\[\begin{equation}KE = \frac{1}{2}mv^2\end{equation}\]
where m represents the mass of the object, and v its speed. This equation shows that the kinetic energy of an object is directly proportional to its mass and the square of its speed. In practical terms, the heavier an object is, or the faster it moves, the more kinetic energy it has.

It is vital to have a good grasp of this concept when considering moving objects in physics. For instance, when an object is in motion and is then brought to a stop, the kinetic energy it had must go somewhere — often it gets converted into other forms of energy, such as thermal energy due to friction.

Net Work Done

The net work done on an object is an important concept that links the forces applied to an object and its subsequent movement. If you push on a book and slide it across a table, you're doing work on the book. This work is technically the energy you're transferring to the object to set it in motion or to change its speed.
\[\begin{equation}W_{net} = \Delta KE\end{equation}\]
where \[\begin{equation}\Delta KE = KE_f - KE_i\end{equation}\]
In this equation, \(W_{net}\) is the net work done on the object, \(KE_f\) is the final kinetic energy, and \[\begin{equation}\Delta KE \end{equation}\]represents the change in kinetic energy. If the net work done is zero, it means that no net external forces have acted on the object to change its kinetic energy. This can happen when forces are balanced and cancel each other out, or when the object is not subjected to any forces at all. In these cases, the speed of the object would remain constant - it would not accelerate or decelerate.

Particle's Speed

The particle's speed is a straightforward measurement indicating how fast the particle is moving. As mentioned earlier, speed is a critical factor in calculating the kinetic energy of an object. When evaluating the effects of net work done on a particle, its speed offers insights into the particle's dynamic behavior.

The equation \[\begin{equation}0 = \frac{1}{2}m(v_f^2 - v_i^2)\end{equation}\]
highlights a situation where the net work done is zero, leading to no change in kinetic energy. In such a context, the final speed (\[\begin{equation} v_f \end{equation}\]) and the initial speed (\[\begin{equation}v_i\end{equation}\]) have a specific relationship. Since the kinetic energy remains unchanged, the final and initial speeds are equal in magnitude, which could represent an object moving at a constant speed or coming to a stop and then moving in the opposite direction with the same speed. This fundamental aspect of a particle's speed is invaluable when predicting and understanding the movement patterns of objects in physics experiments and real-world applications.