# Will the pebble meet with the block according to the given condition?

• Mahfuz_Saim
In summary, Question 1 asks for the correct values of A and ω, while Question 2 asks for the zero initial velocity. By using v= ω√(A^2-x^2), Question 1 gets the value of x. By using t=x/ v cosθ, Question 2 gets the value of x after t second of the projectile. By doing some simple calculation, Question 1 can get the distance from A. However, there is no information about the hight of the block in the question. So, the answer to Question 2 may depend on the height of the block.
Mahfuz_Saim
Homework Statement
In the figure below, AB= 2.5 m and AC= 8 m. The spring is stretched by 0.2 m and then released from rest at t=0 sec so that the spring executes SHM. At the time a pebble is thrown with a velocity V= 11.2 m/s from point C so that the pebble meets the block B at t= 1 sec. The angular frequency of the spring is π/3.
Question 1: Find the velocity of the spring at 1 sec.
Question 2: Will the pebble meet with block B according to the given condition?
Relevant Equations
v= Aω*cos(ωt+δ)
v= ω√(A^2-x^2)
t=x/ v cosθ
Question 1:
I have used v= Aω*cos(ωt+δ) where A= 0.2 m, ω= π/3, t=1 and δ=0. Are the values right in this case? I am confused.

Question 2:
From question 1 I have got the value of V which is 9 m/s. By using v= ω√(A^2-x^2), I have got the value of x. Now, do I need to add it with 2.5(distance between AB) or subtract from 2.5? I am confused. Please explain with the cause.

By using t=x/ v cosθ, I have got the value of x after t second of the projectile. By doing some simple calculation, I can get its distance from A. But there's no information about the hight of the block. So can I say they will meet if the distance from A of them are the same?

Mahfuz_Saim said:
I have used v= Aω*cos(ωt+δ) where A= 0.2 m, ω= π/3, t=1 and δ=0.
What does that give for the velocity at t=0?

Your value of ##\delta## is not correct. It must be such , that at time t=0, the velocity v equals 0.

I think you are supposed to assume that the height of the block is zero. So block and projectile will meet(if they will do) when the projectile hits the ground. So find the time t that the projectile hits the ground, and then find at this time t where exactly the block is. you ll need to write down the equation for x of the block as ##x=A\cos(\omega t+\delta_1)##. (now it will be ##\delta_1=0## i think).

Mahfuz_Saim
@haruspex, do I need to use δ= π/2? Maybe, yes. Why?

@Delta2, there is nothing about the hight in the question. That's more confusing. The range of the projectile is more than the distance of AC. So, maybe if they stay at the same point on t= 1 sec, maybe the answer will be yes. Please let me know what you are thinking.

Mahfuz_Saim said:
@haruspex, do I need to use δ= π/2? Maybe, yes. Why?
To get the zero initial velocity it will have to be +/-π/2. To decide which, consider the displacement at t=0.
Mahfuz_Saim said:
By using v= ω√(A^2-x^2)
Or you could use x=A*sin(ωt+δ) as displacement from B.
Mahfuz_Saim said:
there is nothing about the hight in the question
We are not told the block's width either, so take it as a point mass at ground level.

Okay, but I am not talking about the width. The main problem is with hight. If I consider the block as a point, the question becomes meaningless as the projectile's range is more than AC.

Mahfuz_Saim said:
the projectile's range is more than AC.
And substantially more than the furthest the block ever gets from C. So the answer to that part is...?

Mahfuz_Saim said:
Okay, but I am not talking about the width. The main problem is with hight. If I consider the block as a point, the question becomes meaningless as the projectile's range is more than AC.
Not be my reckoning. Please justify

If I am not wrong, the horizontal range= (v^2*sin2θ)/g. Then it will be 11.08. So, the pebble will land 11.085 m away from C. Is it correct?
Please tell me if I am going in the right way or not. If not, please tell me the right way to solve this question.

sin2##\theta##= 0.5
v=11m/s
##g=10m/s^2##

The angle is 60°. So, sin120°= √3/2. Am I correct?

Oh I guess so! It works out so well with that error.
My apologies this is more interesting (or not!) . Let me regroup.

Mahfuz_Saim
So where is the pebble at t=1s? How high up is it?
Where is the block center at t=1s
I guess you could calculate the minimum size of the block to make a collision at 1s?Otherwise I don't understand the question.

Mahfuz_Saim
By using v= ω√(A^2-x^2), I will get a value of x. Do I need to add with the distance of AB? As the block is stretched, I think I need to add the value of x as I am keeping the value of A positive. Am I right? Please reply. Thank you.

hutchphd said:
I guess you could calculate the minimum size of the block to make a collision at 1s

I would point out that the easier way to find x is to use the explicit harmonic oscillator solution ##x=A\sin(\omega t +\pi /2)##. This sets your origin at the original point B. But your way will give the same result with a little more work.

Mahfuz_Saim
Okay, thank you very much, @hutchphd . This was my second question here and have learned a lot from it.

Did you figure a height for the block?...I would like to know (I didn't work it all the way)

One more thing, do I need to add the value of x with AB or subtract in order to get the distance from A and why? Please answer.

x is measured from the resting center of spring. So draw the system (for yourself) approximately at the time=1s with arrows showing the speed. If the phase and constants are correct ( I think they are ) then that should make it clear. In particular the block should be moving left but not yet to the spring center I believe.

As it is on the left side, they will meet at a point which is 2.4 m away from A. And the minimum height of the block should be 4.7995 m. (If I have not done any mathematical error). Thanks.

I was also wondering where is the center of the block when that happens. it will be to the right of point B so the block will need to be at least wide enough to accommodate...
Good work numbers seem right to me ( but I did not calculate )

## 1. Will the pebble and block collide?

It depends on the given condition. If the condition states that the pebble and block are moving towards each other with enough force, then they will collide. However, if the condition states that they are moving in opposite directions or with very little force, then they may not collide.

## 2. What is the given condition?

The given condition refers to the specific circumstances or parameters that are being used to determine if the pebble and block will meet. This could include factors such as their initial velocities, positions, and any external forces acting on them.

## 3. How can we predict if the pebble and block will meet?

To predict if the pebble and block will meet, we must analyze the given condition and use mathematical equations and principles such as Newton's laws of motion to determine if their paths will intersect at any point.

## 4. Is this a common problem in physics?

Yes, this type of problem is commonly encountered in physics, especially in the study of motion and collisions. It helps to develop critical thinking and problem-solving skills, as well as a deeper understanding of concepts such as velocity, acceleration, and momentum.

## 5. Can we change the given condition to ensure the pebble and block meet?

Yes, by altering the initial conditions such as the velocities or positions of the pebble and block, we can manipulate the given condition to ensure that they will meet. However, this may require a thorough understanding of the underlying principles and equations involved.

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