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There is an example in my textbook worded as follows;

A particle of mass 2kg moves along the positive x axis under the action of a force directed towards the origin. At time t seconds, the displacement of P from O is x metres and P is moving away from O with a speed of v ms^-1. The force has magnitude 40x N. The particle P is also subject to resistive forces of magnitude 16v N.

a) Show that the equation of motion of P is d^2x/dt^2 + 8 dx/dt +20x = 0

I do not understand how the force can be towards the origin and the particle be moving away from the origin?

The diagram in the example shows the origin O to the left of P with the force 40x pointing towards the left , the origin, as well as the resistive force 16v and the acceleration pointing to the right.

I do not understand how you can draw the force of 40x and the resistive force of 16v pointing in the same direction.

If P is moving away from the origin then should the force of 40x be directed to the right, away from O.

However in the algebra they use F=ma with -40x - 16v =2a

Whilst I understand that the -40x implies the force acting in the opposite direction to what is shown in the diagram and therefore to the right and being opposed by the resistive force, this seems to be a very contrived why to end up with a damped harmonic motion when the differential equation is solved.

{given x(0)=0 and x(pi/4)=3e^-pi}

If I start with the 40x acting to the right, then the solving the differential equation does not end up as a harmonic equation.

If you could find a way to explain this situation to a 'bear of very little brain' I would be grateful.

I have tried to think of a practical context where this force and resistance might occur and am thinking of a car accelerating from O at a uniform rate - the 'force' on the accelerator increasing uniformly as x increases and the resitive force being the air resistance which increases with velocity, but I cant see how air resistance, being a multiple of v, would stop the car. Do you know of any practical situation that this model would represent?

Thank you in advance for any assistance you can give