Problem — Ball, spring, friction and flight (4 states)

Problem

A ball ($8\;\mathrm{kg}$) starts from rest compressed against a spring (state A); the spring launches it along an inclined plane with friction (state B); the ball goes up and decelerates until C; then it takes off and continues in free flight up to D. Set up, transition by transition, the energy table: for each step state which columns are needed ($E_\text{kin}$, $E_\text{pot,g}$, $E_\text{pot,m}$, $E_\text{therm}$) and why.

Solution

Guiding principle. We apply energy conservation state by state: the final values of one transition become the initial values of the next. In each table we keep only the columns that actually change or come into play in that transition; the others are zero or constant and can be omitted.

Transition A → B (the spring launches the ball).

• Columns needed: $E_\text{pot,m}$, $E_\text{kin}$, $E_\text{pot,g}$.
• In A the ball is at rest ($E_\text{kin} = 0$) and the spring is compressed: all the energy is $E_\text{pot,m}$. On release, the spring unloads ($E_\text{pot,m} \to 0$) and the energy becomes kinetic; if the ball already starts climbing the ramp, $E_\text{pot,g}$ also appears.
• This transition introduces the column $E_\text{pot,m}$ (elastic energy), which is no longer needed in the following ones (the ball has left the spring).

$E_\text{pot,m}^A = E_\text{kin}^B + E_\text{pot,g}^B$

Transition B → C (climbing the ramp with friction).

• Columns needed: $E_\text{kin}$, $E_\text{pot,g}$, $E_\text{therm}$.
• The ball rises and decelerates: $E_\text{kin}$ converts into $E_\text{pot,g}$ (rise in height) and is partly dissipated by friction.
• This transition adds the column $E_\text{therm}$: friction does negative work and transfers energy into heat, $E_\text{therm} = F_a\cdot\Delta s$. The spring is no longer involved, so $E_\text{pot,m}$ disappears.

$E_\text{kin}^B + E_\text{pot,g}^B = E_\text{kin}^C + E_\text{pot,g}^C + E_\text{therm}$

Transition C → D (take-off and free flight).

• Columns needed: only $E_\text{kin}$ and $E_\text{pot,g}$.
• In flight there is no longer contact with the ramp: no friction (no new $E_\text{therm}$) and no spring. Mechanical energy is purely conserved: the ball exchanges $E_\text{kin}$ and $E_\text{pot,g}$ along the flight parabola.

$E_\text{kin}^C + E_\text{pot,g}^C = E_\text{kin}^D + E_\text{pot,g}^D$

Summary — which columns for each transition:

Transition	$E_\text{kin}$	$E_\text{pot,g}$	$E_\text{pot,m}$	$E_\text{therm}$
A → B (spring)	✓	✓	✓ (introduced)	—
B → C (friction)	✓	✓	—	✓ (added)
C → D (flight)	✓	✓	—	—

$\ev{\text{A}\!\to\!\text{B: introduces } E_\text{pot,m};\ \ \text{B}\!\to\!\text{C: adds } E_\text{therm};\ \ \text{C}\!\to\!\text{D: only } E_\text{kin}+E_\text{pot,g}}$

The heart of the problem is bookkeeping discipline: each column appears only when the corresponding interaction is active, and the energy dissipated into $E_\text{therm}$ never comes back (an irreversible transformation).

Collegamenti

Argomenti: Work and energy Concetti: Kinetic energy · Gravitational potential energy · Elastic potential energy · Thermal energy from friction · Conservation of mechanical energy Competenze: Using the energy table · Bubble-diagram energy balance Metodi: State-by-state energy table Oggetti: Spring · Inclined plane