A planet moves around the sun. At a given point P, it is closed from the sun at a distance d₁ and has a speed v₁ . At another point Q. when it is farthest from the sun at a distance d₂ , its speed will be

The correct option/answer is d₁v₁/d₂.

Let us understand the motion of the planet around the Sun step-by-step.

Step 1: Understanding the Physical Principles
When a planet moves around the Sun under the influence of gravitational force, the gravitational force acts along the line joining the center of the planet and the center of the Sun. This means it is a central force. Since the line of action of the gravitational force passes through the Sun, the torque ($\tau$) acting on the planet about the Sun is zero:
$\tau =0$
According to Newton's laws of rotational motion, when the net external torque acting on a system is zero, its total angular momentum ($L$) remains conserved (constant) throughout the motion.

Step 2: Angular Momentum at Closest and Farthest Points
The angular momentum of a particle of mass $m$ moving with velocity $v$ at a distance $r$ is given by the cross product:
$L=m(r\times v)$
Or, in terms of magnitude:
$L=m·v·r·sin\left(\theta \right)$
where $\theta$ is the angle between the position vector and the velocity vector. At the closest point (perihelion) $P$ and the farthest point (aphelion) $Q$, the velocity vector is perpendicular to the position vector (i.e., $\theta =90°$, and $sin(90°)=1$). Thus, at these two points, the angular momentum simplifies to:
$L=m·v·r$

Step 3: Applying Conservation of Angular Momentum
Let:
- At point $P$ (closest point), distance is $d_1$ and speed is $v_1$.
- At point $Q$ (farthest point), distance is $d_2$ and speed is $v_2$.
- The mass of the planet is $m$.

Applying the law of conservation of angular momentum between points $P$ and $Q$:
$L_P=L_Q$
$m·v_1·d_1=m·v_2·d_2$

Step 4: Solving for the Speed at Point Q
We can divide both sides of the equation by $m$:
$v_1·d_1=v_2·d_2$
Now, solving for the unknown speed $v_2$:
$v_2=\frac{d_1·v_1}{d_2}$
This can be written as:
$v_2=d_1{v}_1/d_2$