What is the optimal velocity for a bouncing ball to pass through a window?

[Oonej]

Homework Statement

Consider there is a window 5 meters from a person. The window is sized 1.21 meters (4 ft) in length and .914 meter (3 ft) wide at a height of 6 meters from the ground.

A person 1.8288 meters tall (6 ft) wants to bounce a ball (COR: 0.789) on the ground such that the ball goes through the window on the first bounce.

If the person bounces the ball with a speed of 25m/s at an angle of -60° from the horizontal at a point in the middle (between him and the wall), calculate and plot the trajectory of the ball.

Will the ball go through the window? If not, change one of the values such that it does. Then find a RANGE of velocities that will meet this condition, while keeping the COR, Distance, and angle the same.

Homework Equations

X0(t) = V0cos(theta)t
Y0(t) = V0sin(theta)t - g/2t^2
V = V0(alpha)^n : alpha = the COR, n = the bounce

The Attempt at a Solution

I formulated an excel sheet to calculate where the ball was at 5 meters along the x direction and discovered that it was too high of a position and would hit above the window. After changing the velocity to 20 m/s it seems to work.

My problem is writing this in a simplistic formula. As well, finding the range of available velocities without having to use an excel spreed sheet for each calculation by "guessing" and trying each available velocity until i find a max and min.

Thanks for any support in this! <3

 

[xiaoB]

COR

Can you tell me what meant by COR ?

 

[Oonej]

The coefficient of restitution (COR), or bounciness[1] of an object is a fractional value representing the ratio of speeds after and before an impact. An object with a COR of 1 collides elastically, while an object with a COR < 1 collides inelastically. For a COR = 0, the object effectively "stops" at the surface with which it collides, not bouncing at all. COR = (relative speed after collision)/(relative speed before collision).

simply multiply this by the velocity after the bounce to get its velocity.

V0*COR = new velocity

or simply V0*COR^n where n is the number of bounces

 

[xiaoB]

The coefficient of restitution (COR), or bounciness[1] of an object is a fractional value representing the ratio of speeds after and before an impact. An object with a COR of 1 collides elastically, while an object with a COR < 1 collides inelastically. For a COR = 0, the object effectively "stops" at the surface with which it collides, not bouncing at all. COR = (relative speed after collision)/(relative speed before collision).

simply multiply this by the velocity after the bounce to get its velocity.

V0*COR = new velocity

or simply V0*COR^n where n is the number of bounces

Thanks ! Oonej.

But from what i have calculated : The ball would go though the window with above 6.13~6.3 m.

But nevermind ! Just ignore me maybe i got mistakes.

 

[Oonej]

Well actually i may have done something incorrectly. can you show me how you did it?

 

[xiaoB]

Well actually i may have done something incorrectly. can you show me how you did it?

Same things as you found V₁=eV_o (normally "e" represents COR) but before you have to find the X₁ distance for projected downward motion is equal to 1m.

Then, you have X₂ distance for bounced upward motion is 4 m and to get the time ~ 0.4s.

Therefore, using vertical equation for y=V₁sin 60⁰t-0.5gt².

That is my solutions.

 

[Oonej]

I must be doing something wrong. How did you solve for the position of X₁ at the first bounce? Can you give a more detailed explanation?

I was given a hint by my professor :

Hint: Look at equations for x(t) and y(t). You could eliminate time from the equations and get an equation for y(x)

thanks so much

 

[xiaoB]

I must be doing something wrong. How did you solve for the position of X₁ at the first bounce? Can you give a more detailed explanation?

I was given a hint by my professor :

Hint: Look at equations for x(t) and y(t). You could eliminate time from the equations and get an equation for y(x)

thanks so much

You so welcome.

You just need to apply these two formulae, X₁=V₀cosθ t and Y₁=V₀sinθ t + 0.5gt² to get your X₁.(Find up the t from Y₁ first)

I have to notice that if you let θ=-60⁰ then your displacement of Y₁ also in negative sign.That is why i shown this Y₁=V₀sinθ t"+"0.5gt² .

 

[Oonej]

I got .08s for the t where it reaches the ground at x₁

 

[xiaoB]

I got .08s for the t where it reaches the ground at x₁

Yup !

 

[xiaoB]

After then you can get X₂=5-1=4 m.

Therefore, from X₂=V₁cosθ t₁ => t₁=X₂/V₁cosθ.(t₁ is for the range X₂ and displacement Y₂ of bounced upward motion).

Finally, t₁ into this formula(Eliminated the t₁):

Y₂=V₁sinθ t₁ - 0.5gt₁². To get your answer.

 

[Oonej]

Now how do you find the range of velocities to make it through the window?

 

[xiaoB]

Will the ball go through the window? If not, change one of the values such that it does. Then find a RANGE of velocities that will meet this condition, while keeping the COR, Distance, and angle the same.

Normally, the Range of X₂=V₁cosθt₁ .So, Range of Velocity is V₁cosθ= X₂/t₁ => V_X2=X₂/t₁. And same things to Range of X₁.Apply this θ=tan^-1(e tan 60⁰). Please check it out the answer. Y₂ is too low.

Very sorry to you.

 

[xiaoB]

Now how do you find the range of velocities to make it through the window?

Let say the maximum height for Y₂ go through the window is equal to 6 m meanwhile the final vertical velocity V_yf=0 for Y₂.

Therefore, apply V²_yf=V²_yi-2gY₂. Because V_iy =V₁sinθ.So,you can get the V₁ after that you can get the V₀.

 

[Oonej]

I can't understand how he did this. Can anyone reword this so I can comprehend it?

The procedure isn't very clear the way it was explained. Or I'm just slow. lol

 

[xiaoB]

I can't understand how he did this. Can anyone reword this so I can comprehend it?

The procedure isn't very clear the way it was explained. Or I'm just slow. lol

This: Here got a mistake please change it ∅=tan^-1(e tan θ).

 

[Oonej]

Can anyone else make sense of this? I greatly appreciate the help Xiaob. But I just don't understand the procedure very well.

I've had my physics teacher look at this and he combined so many formula's that i lost my mind.

If anyone can PLEASE help me figure this out!

Not sure if payment to physics people is allowed on these forums, but I am willing to pay for some help. Consider it tutoring fee's.

Thanks for anyone who helps!

 

[Oonej]

I have discovered that the ball DOES NOT go through the window with the given spec's. But after some brute forcing an excel sheet. I discovered it does go through the window with an angle of -70 degrees at a velocity of 35 m/s

Now I'm stuck at the second part with finding a range.

 

[Oonej]

Ok, more progress...

y2 = x2 tan θ - [ g / ( 2 v1^2 cos^2 θ ) ] * x2^2

using this formula, and changing the degree to 73 (but keeping the velocity at 25 initially which made it through the window. since i was only allowed to change one variable I made it be the angle)

I am getting a few velocities off it seems. minimum calculated out to be 19.1127, but could probably still be a bit slower. and 26.678 for max, but seems it could be a bit faster.

Something I am doing is slightly off. I think it has something to do with the initial x1 value not being calculated correctly.

X2 = 5 - x1, so plugging in 5 won't work, and we don't know what x1 is. Any ideas?

 

[cupid.callin]

This is one long question

i think we can solve like this:

split initial velocity u into into 2 X,Y components ucosθ and usinθ
during collision and falling X component will experience no change
final velocity in Y, v_Y can be found out by: v_Y = e u'
here u' is new value for usinθ after falling 6ft

now for what distance it travels during this first part of motion can be found out by two ways,

I, shorter and using 1 simple formula:
y = xtan\theta - \frac{gx^2}{2u^2 cos^2\theta}
i always prefer not using this formula

II, general way

use y = u_yt + \frac{1}{2}gt^2
and find t
now use X = ucosθ t

hence we find u_X, u_Y and distance from point of initial projection

i will continue rest later

 

[Oonej]

Your way doesn't work, when we have 2 unknowns. We can't find T and V at the same time... or can we?

 

[eczeno]

ok, you have chosen a complicated problem. from what i have read so far your calculations of x_1 = 1.037 m and t_1 =0.083 s are correct, though you have rounded too much which is why my results are slightly different.

the situation now gets a little more complicated. the speed in the y-direction increases slightly during this first part of the motion. I calculate the speed in the y-direction right before the bounce to be -22.4636 m/s and so the total speed of the ball just before the bounce is v_f = 25.7 m/s.

Also, the angle the ball strikes the ground at will no longer be 60^o. the angle I get is now 60.91^o.

Here we have to make an assumption; namely that the ball will obey the law of reflection and bounce away at the same angle it came in. this is not unreasonable, but there are plenty of circumstances in which this would not happen. But, let's make that assumption.

ok, so now the ball bounces. it leaves the floor at an angle of 60.91^o with a total speed of v_2 = 20.2773 m/s. thus, v_x = 9.86 m/s and v_y = 17.72 m/s.

Now, you can calculate how long the ball will take to travel the remaining horizontal distance, and then find the altitude at that time.


In order to find a range of initial velocities, you need to repeat the whole problem, but do not put in a value for v_o. the result will be an expression for the altitude that depends on v_o.

hope this helps.

 

[cupid.callin]

Here we have to make an assumption; namely that the ball will obey the law of reflection and bounce away at the same angle it came in. this is not unreasonable, but there are plenty of circumstances in which this would not happen. But, let's make that assumption.

that is a perfectly wrong assumption

its true only when e=1

 

[cupid.callin]

Your way doesn't work, when we have 2 unknowns. We can't find T and V at the same time... or can we?

which part are you talking about?

 

[Oonej]

yeah the reflecting angle is (COR)(THETA)

 

[Oonej]

Well, from my understanding, you are saying to solve for t, but we don't know what Uy or Ux is...

 

[cupid.callin]

Well, from my understanding, you are saying to solve for t, but we don't know what Uy or Ux is...

isnt u 25m/s?

 

[Oonej]

if you followed the post, we discovered that that the angle -60 degrees, 25 m/s velocity, with the given heights does not work. After further investigation, NO velocity at -60 degree's works. Thus I changed the angle (as it requests in the problem) to -73 degrees. Which works.

Now the goal is to find a range of velocities for the minimum and maximum (through the window) which my lap professor told me would be easiest by combining the formulas y(x) and solving for V but it gets tricky when we don't know what the position is at X1 ( the first bounce) because my formula:

y2 = x2 tan θ - [ g / ( 2 v1^2 cos^2 θ ) ] * x2^2

is requesting X2, but X2 is actually the distance from 5 - X1...

 

[eczeno]

in my opinion, anything we say about the angle of reflection is suspect. there are simply too many factors to account for (is the ball spinning, the coefficient of friction, etc.). so, i think my assumption is a fair one.

i fail to see how the COR has much to do with the angle. from what i understand it is just the ratio of outgoing speed to incoming speed. if the angle depended on that ratio, we would see some strange things indeed. for example, if i throw the ball almost straight down, say 88^o, and the COR is 0.25. then we should expect the ball to shoot off to the right at 22^o above the horizontal. Now, I can make a ball do that, but it would take some serious spin and friction would be absolutely necessary. Maybe i am misunderstanding, but i don't see how COR can affect the angle.

 

[eczeno]

After further investigation, NO velocity at -60 degree's works.

this is not true. there will be a range of velocities for 60 degrees.

 

[Oonej]

that is using your assumption that the angle doesn't change upon the first bounce.

 

[eczeno]

please see my post above. the assumption i make seems the only reasonable one to make.

 

[Oonej]

My understanding, not to discredit yours, is that the ball will encounter a rotation. I have had several people say that it does. I have researched it and can't find any proof.

So its all hearsay... can anyone provide information on this? If it is the case that the ball doesn't change trajectories after impact this problem becomes 10x's easier!

 

[eczeno]

i did say that i was making an assumption, and that there are many circumstances in which the law of reflection will not hold.

if there is friction, and the ball does begin to rotate, which is much more realistic, then the trajectory will probably be altered. But this will not, in my opinion, depend on e. and the equation theta_r = e * theta_i simply cannot be right; it predicts very strange things.

it is not immediately obvious to me how the angle of reflection would change with the introduction of friction, but it certainly depends on information like the coefficient of friction, which is not provided. and so, in order to be able to solve the problem, one must ignore friction and assume the law of reflection holds. i see no other way to solve the problem with the information provided.

cheers

 

[Oonej]

I will assume at this point that the law of reflection holds. and I can probably solve the whole project now :)

 

[cupid.callin]

i did say that i was making an assumption, and that there are many circumstances in which the law of reflection will not hold.

if there is friction, and the ball does begin to rotate, which is much more realistic, then the trajectory will probably be altered. But this will not, in my opinion, depend on e. and the equation theta_r = e * theta_i simply cannot be right; it predicts very strange things.

it is not immediately obvious to me how the angle of reflection would change with the introduction of friction, but it certainly depends on information like the coefficient of friction, which is not provided. and so, in order to be able to solve the problem, one must ignore friction and assume the law of reflection holds. i see no other way to solve the problem with the information provided.

cheers

you should always work with data given to you and ignore what's not

if e is given then that obvoiusly means that v_final is not equal to v_initial
so law of reflection is not followed

and it is possible that we can throw the ball through window if we change velocity

 

[Oonej]

you should always work with data given to you and ignore what's not

if e is given then that obvoiusly means that v_final is not equal to v_initial
so law of reflection is not followed

and it is possible that we can throw the ball through window if we change velocity

Is this ignoring my original assumption that the angle changes upon each bounce? If so, I was aware of that already ;) then someone told me it changes angles...

 

[eczeno]

if e is given then that obvoiusly means that v_final is not equal to v_initial
so law of reflection is not followed

I agree the speeds will be different, but they are speeds, they tell us nothing about direction. if we do not assume the law of reflection, how can we determine the angle of reflection? knowing e tells us nothing about direction, so that will not help us. I fail to see any other way to address the problem of direction than assuming the law of reflection. any deviation from the law of reflection will be due, in most part, to friction. since we don't know the friction, i ignored it.

 

[Oonej]

ah! I figured out why the angle changes!

the COR does not effect the X direction at all. Just the Y. so with the equation of Tan. it is Tan (theta) = y/x well if the y is affected by the COR, then the new angle is infact changed Tan^-1( y(COR) / x )

 

[eczeno]

yes, if that is the definition of COR, then you can use it to find the change in angle.

 

[cupid.callin]

i already told you that

during collision and falling X component will experience no change

 

[Oonej]

Well it just made sense to me when I thought about it :) Not stealing your thunder!

 

[Oonej]

you should always work with data given to you and ignore what's not

if e is given then that obvoiusly means that v_final is not equal to v_initial
so law of reflection is not followed

and it is possible that we can throw the ball through window if we change velocity

what velocity allows you to make the window? Because I tried a lot of different combination

 

[eczeno]

sorry cupid, i totally missed that as well. understanding the proper definition of a term is very helpful. :)

 

[Oonej]

I made a chart with a very HIGH velocity to see the impact. and with an extremely high velocity the ball was still landing under the window. The angle change is what screwed it up.

 

[Oonej]

Ok there was a correction to this problem. My TA assigned this project with an error.

We are to assume the ball does not change angle after each bounce, the COR only affects the Velocity.

If you guys could please shed some light...

I took Y₀ formula, combined it with the x₀ formula, and made a y9x) formula. after which I had a Y₁(x) and Y₂(x) and put in and x2 i replaced with 5 - x1 and had an enormously large equation that took forever to solve... and it gave me a number that was ALMOST correct... I got v1 to equal ~15.5 and then divided by my COR² and got 24.898 ... the correct answer is somewhere around that, between 23 and 24 for the min... the max is huge around 800-900ish ( i think ) but hoping there is a an easier solution to this problem ...

Thanks for any help guys! <3

 

[Oonej]

After brute forcing an excel sheet with numbers, i came to the conclusion that there are numerous min and max's ...

for instance, a min of 24

a max of ~191, ~780 etc seem to all work, unless i did something wrong in my formulas...
but like a velocity > 191 fails, but when its around 700ish, it works again...

how can this be?