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Step-by-Step Solution
Step 1: Identify the Physical Principle
In planetary motion, when a planet revolves around the Sun, its angular momentum about the Sun remains constant (assuming no external torque). This principle of conservation of angular momentum forms the basis to relate the speed at different points in the orbit.
Step 2: Set Up the Angular Momentum Conservation Equation
Angular momentum $L$ is given by the product of linear momentum ($m v$, where $m$ is the mass of the planet and $v$ is its linear velocity) and the perpendicular distance from the axis of rotation ($r$). For a planet of mass $m$, the angular momentum about the Sun is:
$L = m \, v \, r$
Since $m$ is constant, the conservation of angular momentum implies:
$v_{\text{min}} \, x_{\text{max}} = v_{\text{max}} \, x_{\text{min}}$
where
$v_{\text{min}}$ is the minimum orbital speed of the planet (given as $v_0$),
$v_{\text{max}}$ is the maximum orbital speed of the planet,
$x_{\text{min}}$ is the minimum distance (perihelion) from the Sun (given as $x_1$),
$x_{\text{max}}$ is the maximum distance (aphelion) from the Sun (given as $x_2$).
Therefore, mathematically:
$v_0 \, x_2 = v_1 \, x_1$
where $v_1$ is the maximum speed we need to find.
Step 3: Solve for the Maximum Speed
From the equation above, solve for $v_1$:
$v_1 = \frac{v_0 \, x_2}{x_1}$
Step 4: Final Answer
Hence, the maximum speed of the planet is:
$v_1 = \frac{v_0 \, x_2}{x_1}$
