Question

If the net work done by external forces on a particle is zero, which of the following statements about the particle must be true? (a) Its velocity is zero. (b) Its velocity is decreased. (c) Its velocity is unchanged. (d) Its speed is unchanged. (e) More information is needed.

Short answer

The correct option is (d): its speed is unchanged.

Step-by-step solution

The work-energy theorem

According to the work-energy theorem, when the external force act on the system, the total work done on the system is equal to the change in the system's kinetic energy.

Let’s assume the initial velocity of the particle as $v_{\mathrm{i}}$, the final energy of the particle as $v_f$, and the work done by the external force is W. Then the work-energy theorem can be expressed as follows:

$\mathrm{W}=\frac{1}{2}{\mathrm{mv}}_{\mathrm{i}}^2-\frac{1}{2}{\mathrm{mv}}_{\mathrm{f}}^2$ (1)

Here, m is the mass of the particle.

Determine the correct statement among the given statements

It is given that the net work done on the particle is zero.

Substitute 0 for W into equation (1).

$\begin{array}{l}0=\frac{1}{2}m{v}_i^2-\frac{1}{2}m{v}_f^2\\ \frac{1}{2}m{v}_i^2=\frac{1}{2}m{v}_f^2\\ v_i^2=v_f^2\\ v_i=v_f\end{array}$

Therefore, the final speed of the particle is equal to the initial speed of the particle. The work-energy theorem gives information about the particle's speed but not the direction of the particle.

Hence, the speed of the particle is unchanged.