Conservation of Mechanical Energy — Question 9

Question 9

A conserved scalar leaves velocity direction undetermined

A 2 kg particle has constant mechanical energy E=25 J in a conservative planar field. At point P the potential is 9 J. Its x velocity there is 2 m/s.

1. Find kinetic energy and speed at P.

2. Find the possible y velocity components.

3. Can conservation of mechanical energy alone choose between those possibilities or determine arrival time?

Question 9 - Solution

A conserved scalar leaves velocity direction undetermined

(a) Subtract the potential from the total.

At P, the kinetic energy is the energy gap: $$\boxed{K=25-9=16\ \mathrm J},\qquad\boxed{v=\sqrt{2K/m}=\sqrt{16}=4\ \mathrm{m/s}}.$$

(b) Use the component relation.

Resolve the speed into components: $$v_x^2+v_y^2=v^2\Rightarrow v_y^2=16-2^2=12,$$ $$\boxed{v_y=\pm2\sqrt3\ \mathrm{m/s}}.$$

(c) Retain the missing trajectory information.

The scalar constraint $$\tfrac12m(v_x^2+v_y^2)+U=E$$ is unchanged by $v_y\mapsto-v_y$. Initial direction and the force-driven trajectory are needed to select the sign and determine arrival time.