Discover the Height for Perfect Ball Hooping with Our Grade Lab Homework Help"

[Physics_B]

Homework Statement

A ball is dropped down a hotwheels ramp attached to the end of a table. The ball is then propelled off the table and aimed at a pole with a "hoop". The ball must fall into the hoop. Find the height of the pole to make this possible.

My instructor never gave us any numbers as he is in favor of working in variables.

Homework Equations

The system has 3 "points" at which equations are given. Point A, the top of the ramp, equals Ug, which equal mgh. Point B, the bottom of the ramp, equals Ug+KE, which equals mgh(b)+(.5)(m)(v(b)^2). And lastly, Point C, which equals mgh(c)+(.5)(m)(v(c)^2). These equations equal each other.

The conservation of energy equation is also used. mgh+1/2mv^2=mgh+1/2mv^2

The Attempt at a Solution

I used pythagorean theorem to find the velocity at Point C.
v(x)^2+v(cy)^2 =v(c)^2
v(cy)= g*deltax/v(x)

v(c)=2v(x)+g*deltax

then i plugged it back into the conservation of energy equation, canceled out the mass as dividing it eliminated it, and got

(1/2v(b)^2+gh(b)-1/2(g*deltax))/g=V(c)

V at point c being the final velocity.This is my first physics course and I'm not sure if I applied the information right, and I would appreciate any help I can receive. :)

 

[haruspex]

Sorry, but I find your description of the set-up quite unintelligible. I don't know what a hotwheels ramp looks like. Does the ball go down a longish ramp then up a short one to give it some upward velocity? If so, say what variable names you are giving to the initial height, the height at launch point, the height of the hoop, the horizontal distance from launch point to hoop, and the radius of the ball. Add any further variable names you want, so long as you make clear what they refer to.
Having done that, write out some equations.

 

[Simon Bridge]

All the hot-wheels ramps I've seen launch horizontally.

When the ball is launched horizontally - it starts falling: what determines how far the ball falls by the time it reaches the pole?
[edit] Actually - rereading - that seems to be what you did pretty much. You didn't need the pythagoras bit though.

 

[CWatters]

A diagram would certainly help but it's not impossible without one...

Lets assume the track launches the ball at an angle to the horizontal. The problem is then in two parts...

1) Apply conservation of energy to write an equation for the velocity of the ball as it leaves the ramp.

2) The ball will leave the track at an angle to the horizontal (equal to that of the ramp) with the velocity calculated above. You can resolve that into horizontal and vertical components. It's then a standard projectile problem. If necessary you can work out the distance it will go before hitting the floor, the max height it will reach, time of flight etc. The height of the hoop will depend on how far it is from the launch ramp. This can be done even if the ramp is horizontal or vertical.

 

[Nickie]

Great job on applying the conservation of energy equation and using the Pythagorean theorem to find the velocity at Point C! Your approach seems to be on the right track.

To find the height of the pole for perfect ball hooping, we can set up an equation using the fact that the ball must have enough horizontal velocity to reach the pole and enough vertical velocity to fall into the hoop. This means that the horizontal distance traveled by the ball (deltax) must equal the height of the pole (h).

Using your equation for the velocity at Point C, we can set up the following equation:

h = (1/2v(b)^2+gh(b)-1/2(g*deltax))/g

Solving for deltax, we get:

deltax = (v(b)^2 + 2gh(b))/g

This is the minimum horizontal distance the ball must travel in order to reach the pole and fall into the hoop. Therefore, the height of the pole for perfect ball hooping is equal to this horizontal distance.

I hope this helps! Keep up the good work in your physics course.