What is a differential equation?

[KAV]

Hello all,

I have been checking out practice exams for mechanics. One problem described a skier of mass m sliding down a frictionless hill that makes an angle θ with the horizontal. The skier experiences a velocity dependent drag force as a result of air resistance defined by the equation F=-bv (v is the velocity of the skier and b is a positive constant).

Part (b) of the problem requests a differential equation that can be used to solve for the velocity of the skier as a function of time.

What does this mean? I am not currently enrolled in a calculus course. Is there any way I can learn to work with this terminology without taking calculus yet?

-KAV

 

[mgb_phys]

A differential equation is one where one of the values varies with one of the others.
So in the case of the skier the drag depends on their speed, but their speed depends on the drag - a problem!
Some equations you can solve using calculus, there are well known solutions to many of these equations, others you have to solve numerically.

If you haven't done calculus you can still do this with a computer.
iIn your example - imagine the skier at some speed, work out the drag for that speed, now a short time later work out how much they would have accelerated due to gravity and how much they would have slowed due to the drag - you now know have their new speed. You can repeat this to get the speed as a function of time.

All calculus is - is a way of mathematicaly dealing with things that change by imagining that they are constant for some very small time and then adding up these small times.
Unfortunately wikipedia goes into a lot of technical details and I don't know any simple online tutorials that don't use the complex looking notation that comes with calculus.

Ps. I will ask for this to be moved to homework, I know you aren't asking a specific homework question, but more people will see it there and may have some helpful links.

 

[Astronuc]

The response of a mass to a force is given by the product of mass and acceleration, and acceleration can be written at the first derivative of velocity with respect to time (dv/dt), or the second derivative of displacement with respect to time (d²x/dt²). Remember, v = dx/dt (where v and x are colinear, here assuming Cartesian coordinates).

Ref: http://hyperphysics.phy-astr.gsu.edu/hbase/acca.html

One writes a force balance equation for the mass, in which one equates the response to sum of the forces. In the problem here, one must identify the force pulling the skier down the slope, and the force opposing the skier's motion down the slope.

 

[KAV]

Thanks for the help everyone. It is very much appreciated!

There's no need to have this question moved to homework, as I already asked it there...I realized a moment too late that there was a more appropriate section for this. Apologies to those that keep the order.

-Kevin