Kinetic Energy — Question 7

Question 7

An energy measurement fixes speed but not a missing sign

Use classical translational kinetic energy $K=\tfrac12 m|\vec v|^2$. A 4 kg particle has kinetic energy 50 J, with $v_x=3$ m/s and $v_y=0$. The z component is unknown.

1. Find all possible $v_z$ values.

2. What additional observation distinguishes the two velocities?

3. An observer claims the same particle has $v_x=6$ m/s with unchanged mass and energy. Is any real $v_z$ compatible?

Question 7 - Solution

An energy measurement fixes speed but not a missing sign

(a) Recover the squared speed.

The governing relation is $$\boxed{v_z^2=\frac{2K}{m}-v_x^2-v_y^2\geq0}$$

$$v^2=2K/m=25\ \mathrm{m^2/s^2}.$$ Thus $$v_z^2=25-9=16,$$ giving $$v_z=\pm4\ \mathrm{m/s}.$$

(b) Measure a directional property.

Knowing whether the particle moves toward increasing or decreasing z selects +4 or $-4\ \mathrm{m/s}$. Energy alone is insensitive to that sign.

(c) Require a nonnegative squared remainder.

The claimed x component already gives $$v_x^2=36>25.$$ It would require $$v_z^2=-11$$ even with $v_y=0$, so the observations cannot describe one classical particle.