How do you find the slope of a height versus distance graph? (details followed)?

Question

For the theory part of my lab, we are to find the theoretical slope of the height at which a Hotwheels car rolls down a ramp versus the distance the car travels horizontally after it reaches the bottom. I have the actual slope from a graph I constructed, but I don't know how to find it theoretically. Someone told me it was height = the coefficient of friction times distance, but I don't know if it is right. If someone could just help me find an equation, that's all I need. Thanks in advance.

***EYEONTHESCREEN***

You said that distance = height divided by coefficient of friction. We don't have the coefficient of friction, so how am I going to calculate the theoretical slope?

Answer 1

"we are to find the theoretical slope of the height at which a Hotwheels car rolls down a ramp versus the distance the car travels horizontally after it reaches the bottom."

so calculate the potential energy first, the height from which the car rolls down = mgh

and then at the bottom poten = 0

0.5 mv^2 = mgh

0.5 v^2 = gh

but final v = 0
intial v.. find it using the above formula

v(f)^2-v(i)^2 = 2as

isolate s, which would be the distance the car travelled.

so s= (0-v(i)^2)/a
or s = (-(2gh))/a

find 'a' now

Force of friction once the cars starting going horizontally..
F = ma
however Fric = Fn (u) u=friction constant
=mg (u)
so
a = g(u)

put a value above there
s = (-(2gh))/(g(u)

s = (2h)/u

so, i came up with this relationship
h = height, u friction coefficient and s = distance

try to understand how i reached here..rather than copying that formula.. because it can be wrong
and secondly.. u would need to explain in theory

=============
now, see it's easy to get slope in there..
more the slope.. more is the height

h= slope * horizontal distance of the ramp.. or whatever u would call..

Answer 2

Conservation of energy is invoked here. PE = KE; where PE = potential energy and KE = kinetic energy. The thing to remember is that when your hotwheels is lifted to a height (h) on the ramp, it has PE = mgh potential energy; m = mass of the car and g = 9.81 m/sec^2.

When the hotwheels is released that PE is converted into KE, kinetic energy, because of the conservation of energy. That is, if PE is lost (because the car is losing height as it rolls), that energy is conserved by becoming KE. This is an important concept (law), energy is neither created nor destroyed, it is only converted.

OK, at the bottom of the ramp, what is the PE? If you answered zero, you get a gold star. PE = mgh = 0 because h = 0 at the bottom of the ramp. So where did all that PE go...into KE = 1/2 mv^2.

Since all the PE went into all the KE, we can write PE = KE; so that mgh = 1/2 mv^2; where m = mass of the hotwheels, g = 9.81 m/sec^2 the acceleration due to gravity at Earth's surface, and v = the velocity of the now speeding hotwheels at the bottom of the ramp.

Now here's where k = the coefficient of kinetic friction (rolling friction) comes in. If your hotwheels encountered no friction on the ramp or flat surface when it reached bottom, it would go forever. There would be no force to stop it once it got going.

That's clearly unrealistic, we all know the car will eventually stop. So there must be friction. In fact, there will be kinetic or rolling friction on the ramp as well as the flat part at the bottom. You need to fold the friction forces into the conservation of energy equations. How to do that?

KE = W = Fd = kNd = kmgd = PE = mgh; this string of equations say "kinetic energy is equal to work, which is equal to friction force acting over a distance, which is equal to potential energy (by conservation)."

Because each term is equal to all the others, we can set kmgd = mgh; so that kd = h and d = h/k

AH HA, the distance (d) your hot wheels will roll after hitting the bottom of the ramp is nothing more or less than the height (h) it started from divided by the coefficient (k) of kinetic friction. (I assumed a frictionless ramp.)

W = Fd is the classic equation for work (W), but work and energy are the same thing; so we simply equate the amount of work done to the amount of kinetic energy the car has to convert into heat etc. in order to stop because of friction. F is the friction force, which is F = kN = kmg; where N = the normal weight of the car perpendicular to the flat surface the car is rolling out on.

If you do not assume a frictionless ramp, KE < PE rather than equal. This results because some of that PE will be converted to heat etc. due to the ramp's friction. As a consequence d < h/k would result and the car would not roll out as far as when a frictionless ramp was assumed.

And, to explicitly answer your question. 1/k is the slope of the curve defined by d = (1/k)h. Notice that, when k = 0, d => infinity, which means the hotwheels would go on forever without friction (k = 0).

Answer 3

If your the person like me to try out experiment, then i would like to suggest you one new technic to find the slope. Use this way.

First find the speed to point which you want to find the slope and duration of travel. (vi to vf) (t)

find the Vavg = (vf-vi)/2

then the distance x = Vavg/t

Because you only going to calculate the slope distance only. So don't need to consider the friction factor. But in practical we have to consdier the friction factor.

i don't know wheather i am correct or not. But this is basic concepts

Answer 4

by using the formula y^2-y^1/x^2-x^1