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, the magnitude of the particle's speed does not change. The particle's final speed is either equal to or opposite in direction to its initial speed.

Step-by-step solution

Recall the Work-Energy Theorem

The Work-Energy Theorem states that the work done on an object is equal to the change in its kinetic energy: W = ΔK where W is the work done, and ΔK is the change in kinetic energy.

Express the Kinetic Energy in Terms of Speed

The kinetic energy (K) of an object can be expressed in terms of its mass (m) and speed (v) as follows: K = \dfrac{1}{2}mv^2 Thus, the change in kinetic energy can be expressed as: ΔK = K_f - K_i= (\dfrac{1}{2}mv_f^2 )- (\dfrac{1}{2}mv_i^2 ) where K_f is the final kinetic energy, K_i is the initial kinetic energy, v_f is the final speed, and v_i is the initial speed.

Set the Work Done to Zero

Given that the net work done on the particle is zero, we set W = 0 and substitute the expression for ΔK: 0 = (\dfrac{1}{2}mv_f^2 )- (\dfrac{1}{2}mv_i^2 )

Solve for the Relationship between Speeds

We can now solve for the relationship between the final and initial speeds: 0 = \dfrac{1}{2}m(v_f^2 - v_i^2 ) Since the mass cannot be zero, we have: 0 = v_f^2 - v_i^2 v_f^2 = v_i^2 Taking the square root of both sides, we get: v_f = ±v_i

Interpret the Result

If the net work done on a particle is zero, then the particle's final speed (v_f) is either equal to or opposite in direction to its initial speed (v_i). The magnitude of the particle's speed does not change when the net work done on it is zero.