A body of mass m starts sliding from the top of an inclined…

A body of mass $m$ starts sliding from the top of an inclined plane of height $h$. Find its kinetic energy at the start, halfway along the path and at the bottom. Neglect friction.

Explanation of Method

Kinetic energy along a frictionless incline.

Step 1: Energy conservation

The total energy stays $mgh$; the kinetic energy is the potential energy released so far. Halfway along the incline the body has descended $h/2$.

\[\begin{gathered} E_{k,\text{top}} = 0, \quad E_{k,\text{middle}} = \tfrac12 mgh \\ E_{k,\text{bottom}} = mgh \end{gathered}\]

$E_k = 0$ $\tfrac12 mgh$ $mgh$ $A$ $M$ $B$ $E = mgh$ $h$

potential energy
kinetic energy

Without friction the total energy stays $mgh$; the kinetic energy (red) is whatever part of it has been released by descending.

Result

\[\boxed{\,\begin{gathered} E_{k,\text{top}} = 0, \quad E_{k,\text{middle}} = \tfrac12 mgh \\ E_{k,\text{bottom}} = mgh \end{gathered}\,}\]

Remark

Check. The original results are correct. They do not depend on the angle of the incline. The speed at the bottom is $\sqrt{2gh}$, and halfway it is $\sqrt{gh}$, about 71% of the final speed.