a satellite in force free space sweeps stationary interplanetary dust at a rate of dM/dt=alpha v, where M is mass and v is speed of satellite and alpha is a constant. the tangential acceleration of satellite
F=-vdm/dt=v@v=v²@
a=F/m=v²@/m
To calculate the tangential acceleration of the satellite, we need to analyze the forces acting on it. In force-free space with no external forces, the only force acting on the satellite is the one caused by the sweeping of interplanetary dust.
We are given that the rate of change of mass of the satellite, dM/dt, is equal to alpha multiplied by v, where M is the mass of the satellite and v is its speed. The constant alpha represents the effect of the sweeping of interplanetary dust on the satellite's mass.
Since the satellite's mass is changing, we can apply Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma).
In this case, the mass is changing with time, so we rewrite the equation for force as:
F = M * dv/dt + v * dM/dt
Where dv/dt is the rate of change of velocity of the satellite. However, since the satellite is moving in a straight line tangentially to its orbit, there is no change in its velocity with time (dv/dt = 0). Therefore, the equation simplifies to:
F = v * dM/dt
Now, we substitute the given value of dM/dt with alpha * v:
F = v * alpha * v
Since the force acting on the satellite is equal to the product of its mass (M) and its acceleration (a), we can equate the above expression to M * a:
M * a = v * alpha * v
Therefore, the tangential acceleration (a) can be calculated as:
a = (alpha * v^2) / M
So, the tangential acceleration of the satellite in force-free space is given by:
a = (alpha * v^2) / M