Question 3

A satellite moving with velocity $$v$$ in a force free space collects stationary interplanetary dust at a rate of $$\frac{dM}{dt} = \alpha v$$ where $$M$$ is the mass (of satellite + dust) at that instant. The instantaneous acceleration of the satellite is

Solution

Solution & Explanation

1. Apply the Variable Mass System Equation

For a system with a changing mass configuration, the generalized thrust equation derived from Newton's Second Law is given by:

$$F_{\text{ext}} = M\frac{dv}{dt} - v_{\text{rel}}\frac{dM}{dt}$$

Where:

$$F_{\text{ext}}$$ = Net external force acting on the system.
$$v_{\text{rel}}$$ = Velocity of the incoming mass relative to the main body ($$v_{\text{dust}} - v_{\text{satellite}}$$).
$$\frac{dM}{dt}$$ = Rate at which mass is added to the system.

2. Substitute System Parameters

Let us evaluate the specific physical conditions given in the problem statement:

The system is traveling through a force-free space, which means the net external force is zero:
$$F_{\text{ext}} = 0$$
The interplanetary dust is initially stationary ($$v_{\text{dust}} = 0$$). Therefore, its relative velocity with respect to the moving satellite is:
$$v_{\text{rel}} = v_{\text{dust}} - v = 0 - v = -v$$

Substituting these parameters into our variable mass framework yields:

$$0 = M\frac{dv}{dt} - (-v)\frac{dM}{dt}$$

$$M\frac{dv}{dt} + v\frac{dM}{dt} = 0 \implies M\frac{dv}{dt} = -v\frac{dM}{dt}$$

3. Compute Instantaneous Acceleration ($$a$$)

Isolate the acceleration parameter ($$a = \frac{dv}{dt}$$) from the dynamic equation:

$$a = -\frac{v}{M}\frac{dM}{dt}$$

We are given that the dust accumulation rate scales directly with velocity as $$\frac{dM}{dt} = \alpha v$$. Substituting this rate into the equation gives:

$$a = -\frac{v}{M}(\alpha v)$$

$$a = -\frac{\alpha v^2}{M}$$

Concept Check: The negative sign physically represents a retarding thrust force. As the satellite collides with and absorbs stationary dust particles, it must continuously transfer momentum to accelerate them up to its own speed, which consequently causes the satellite itself to decelerate.