Solving for k, we get k = 2mgd(sinθ)/x2.

[flyingpig]

Homework Statement

A car with mass m rolls d down a frictionless $$\theta^0$$ degree incline. If there is a horizontal spring at the end of the incline, what spring constant is required to stop the car in a distance of x?

The Attempt at a Solution

[PLAIN]http://img4.imageshack.us/img4/8837/unledxrq.jpg

I am honestly very embarrassed that I can't do these basica problems properly and I am a rising sophomore.

I tried

$$\sum W = \Delta K$$

$$mg(d\sin\theta) - \int_{x = 0}^{x = x} kx dx = -\frac{1}{2}mv_0^2$$

I thought that at the top before it collides that

$$-\frac{1}{2}mv_0^2 = -mg(dsin\theta)$$$$mg(d\sin\theta) - \int_{x = 0}^{x = x} kx dx = -mg(d\sin\theta)$$
$$\int_{x = 0}^{x = x} kx dx = 2mg(d\sin\theta)$$$$k = \frac{4mgd\sin\theta}{x^2}$$

Apparently this is wrong and the answer is just mgh = 0.5kx^2.

I decided to not use LaTeX for the correct answer because I am upset.

If assume initial velocity is 0 I get it right
$$k = \frac{2mgd\sin\theta}{x^2}$$

Why? How do I know it started with 0 velocity?

 

[tiny-tim]

hi flyingpig!

$$mg(d\sin\theta) - \int_{x = 0}^{x = x} kx dx = -\frac{1}{2}mv_0^2$$

yes

(except you really ought to use a different variable inside the ∫, maybe x' or y)

I thought that at the top before it collides that

$$-\frac{1}{2}mv_0^2 = -mg(dsin\theta)$$

i don't understand where this comes from

 

[flyingpig]

It came from at the top of the ramp.

 

[PeterO]

Why? How do I know it started with 0 velocity?

This is the simple part.

Read the question and look for the following:

An object falls, an object is dropped, a car rolls down a hill, a block slides down a slope, ...

In all these cases the object is starting with zero velocity.

When the question starts

An object is thrown, A projectile is lauched, or similar, it is NOT starting with a velocity of Zero.

Actually "an object is dropped/falls" can be tricky, since it can sometimes be dropped/fall from a moving point - but then you will realize it is moving so it shouldn't be a problem.
Examples could be:
You dropped a book while in a lift traveling up at 2 m/s
A bomb falls from an airplane traveling at ...
A bag of nuts falls out of a roller coaster when it speeds over a crest ...

Good luck

Peter

 

[SammyS]

At the top of the ramp, the car has PE = mgd(sinθ) and KE = 0.

At the point where the spring has stopped the car, the car has PE = (1/2)k(x²) and KE = 0.

Therefore,

mgd(sinθ) = (1/2)k(x²) .​