Mechanics: finding the speed and positional vectors of a particle.

Any help?!?!

Acceleration is not constant, so suvat does not apply.

You need to work the situation from scratch. Work out the acceleration and start integrating.

Note: There is no force in the +ve x direction, only the one in the -ve x direction.

Okay well if that's the case then I think this is a kind of mechanics that I have not encountered before. If I remember correctly you differentiate acceleration to get speed and then again to get distance but I'm not sure how I would go about getting the acceleration.

but I'm not sure how I would go about getting the acceleration.

This is rather a basic mechanics question, even for A-level, let along uni, and your notes should cover it. I'd suggest getting yourself familiar with them, if you're not already.

But to start you need to be using F=ma, which you should be familiar with.

Suppose that a boat moves in a straight line with speed v through water with resistance proportional to the square of its speed v with initial speed v_0 and initial position x_0. Find its speed v(t) and position x(t) and conclude that the boat travels an infinite distance, i.e. x(t) tends to infinity as t tends to infinity.

Just rewrite what you have been given to form equations for the variables.

e.g. if the resistance is proportional to the square of speed, this means that the rate of change of speed (i.e. acceleration)
satisfies

$\frac{dv}{dt} = \lambda v^2$

Ah okay so as speed is equal to the integration of acceleration, this would mean that $v={\lambda}v^2t+c$ and $x=({\lambda}v^2t^2)/2 + ct + d$. Where c and d are constants. So, where do V_0 and x_0 come in to things?

Ah okay so as speed is equal to the integration of acceleration, this would mean that $v={\lambda}v^2t+c$ and $x=({\lambda}v^2t^2)/2 + ct + d$. Where c and d are constants. So, where do V_0 and x_0 come in to things?

You probably ought to go back and brush up on the basics like the definitions of speed, acceleration, integration, differentiation, basic seperable ODEs and proportionality before tackling this question. Otherwise, we might as well be talking to you in Russian for all you understand of anything. You will have to apply some interpretation too since at some point you will have to decide whether $\lambda$ should be positive or negative for instance.

Velocity is a function of time and acceleration is the rate of change of velocity with respect to time.

If you like, rewrite what I wrote as

$\frac{dv(t)}{dt} = \lambda v^2(t)$

This is a seperable ODE. As is usual with indefinite integrals, the set of solutions will parameterised by a constant. As is also usual, specifying an initial value of $v(t)$ i.e. a value $v_0 = v(0)$ specifies a particular solution inside that set.

You weren't given a numerical value for the initial value v_0.

Therefore all you can do is express the constant of integration in terms of it.

You weren't given a numerical value for the initial value v_0.

Therefore all you can do is express the constant of integration in terms of it.