Collisions between two balls in 2d-motion

In summary, the conversation discusses a problem where a ball A is dropped from rest and another ball B is pushed away with a starting speed v_0. The question asks for the angle α needed for B to collide with A. The equations of motion are presented and the condition for collision is discussed. It is determined that in order for the angle to be 45 degrees, the velocity of B must approach infinity. Another approach using relative velocity is suggested to find the angle of collision.
  • #1
zeralda21
119
1

Homework Statement



A ball A is dropped from rest from a
height of 2.0 m above the floor. Meanwhile that ball A is released, an other ball B is pushed away with the starting speed v_0 from the position shown in Figure. Which angle α is needed for B to collide with ball A?

6B3IqsR.png


Homework Equations



Equations of motion which I present below.

The Attempt at a Solution


Yes, one solution is that it does depend on v_0 but a second solution is at an angle 45 degrees, which I want to find.

I lay out the equations in the x/y direction and then determine A/B equations of motion from these.

X: a(t)=0 Y: a(t)=-g
v(t)=v_0cosα v(t)= -gt+v_0sinα
x(t)=v_0cosαt+x_0 y(t)= -1/2gt^2+v_0sinα+s_0

The equations x(t) and y(t) for A and B are now determined;

A: x_1(t) = 1 I have decided that x_0 refers to x_0=0 for B.
y_1(t) = 2-1/2gt^2

B: x_2(t) = v_0cosαt
y_2(t) = 1-1/2gt^2+v_0sinαt

For collision we need that x_1(t)+y_(1)t = x_2(t)+y_2(t) Right?

But that leads to v_0t(cosα+sinα)=v_0t(√2sin(α+π/4))=2 which clearly is impossible for any α...
 
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  • #2
zeralda21 said:
For collision we need that x_1(t)+y_(1)t = x_2(t)+y_2(t) Right?
You need to rethink the condition for collision.
 
  • #3
DrClaude said:
You need to rethink the condition for collision.

Well at the point of collision, the x-coordinate and y-coordinate must be equal for A and B at the same time. Hence;

[tex]v_0cosα=1[/tex]
[tex]1-1/2gt^2+v_0tsinα=2-1/2gt^2 ⇔ v_0tsinα=1[/tex]

Thus we end up with the system of equations: [tex]\left\{\begin{matrix}
v_0\cos\alpha=1 & \\
v_0t\sin\alpha=1 &
\end{matrix}\right.[/tex] which clearly does not have solutions.
 
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  • #4
zeralda21 said:
Well at the point of collision, the x-coordinate and y-coordinate must be equal for A and B at the same time. Hence;

[tex]v_0cosα=1[/tex]
You're missing a ##t## in there, from which you can get ##t_c##, the time at collision.
 
  • #5
Actually, 45 degrees does not seem reasonable at all. Consider the right-angled triangle BAC where C is the point 1 meter below A. It is a triangle with sides 1,1,√2 and angle ABC is 45 degrees. When ball A starts moving downward, the side AC gets smaller and hence the angle gets smaller.

In order for the angle to be 45 degrees, the ball B must hit A immediately and thus requiring that v_0→∞. Is this analysis correct?

Edit;

You're right;
[tex]\left\{\begin{matrix}
v_0t\cos\alpha=1 & \\
v_0t\sin\alpha=1 &
\end{matrix}\right.[/tex] leads to [tex]\tan\alpha=1 \Rightarrow \alpha=\frac{\pi}{4}[/tex]

In pure interest; What does actually x_1(t)+y_1(t)=x_2(t)+y_2(t) mean geometrically?(In the case that there exist a solution). Yes, totally forgot gravity. Thanks a lot DrClaude.
 
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  • #6
zeralda21 said:
In order for the angle to be 45 degrees, the ball B must hit A immediately and thus requiring that v_0→∞. Is this analysis correct?

You're forgetting gravity.
 
  • #7
No need for so much analysis! Consider the relative motion of B with respect to A(consider A to be at rest). The relative acceleration is zero(as both are under influence of gravity). This implies that the path of B as seen by a will be a straight line also it will have constant velocity. The collision is therefore only possible if the velocity vector of B passes through initial position of A giving angle of 45 degrees directly!
 
  • #8
consciousness said:
No need for so much analysis! Consider the relative motion of B with respect to A(consider A to be at rest). The relative acceleration is zero(as both are under influence of gravity). This implies that the path of B as seen by a will be a straight line also it will have constant velocity. The collision is therefore only possible if the velocity vector of B passes through initial position of A giving angle of 45 degrees directly!

Nice! I like this approach better.
 
  • #9
zeralda21 said:
Nice! I like this approach better.

This is one of the examples where relative velocity is insanely advantageous.
 

FAQ: Collisions between two balls in 2d-motion

1. What is the difference between an elastic and an inelastic collision between two balls?

In an elastic collision, both kinetic energy and momentum are conserved, meaning that the balls bounce off each other with no loss of energy. In an inelastic collision, some kinetic energy is lost and the balls stick together after impact.

2. How does the mass of the balls affect the outcome of a collision?

The mass of the balls affects the outcome of a collision by determining how much momentum each ball has. In a collision, the total momentum of the system is conserved, so a heavier ball will transfer more momentum to a lighter ball.

3. Can the direction of motion affect the outcome of a collision between two balls?

Yes, the direction of motion can affect the outcome of a collision. If the balls are moving in the same direction, the collision will be less elastic than if they are moving towards each other. This is because their velocities are already aligned, so less energy is transferred during the collision.

4. How does the coefficient of restitution impact the behavior of a collision between two balls?

The coefficient of restitution is a measure of the elasticity of a collision. It affects the behavior of a collision by determining how much energy is transferred between the balls. A higher coefficient of restitution means a more elastic collision, while a lower coefficient of restitution means a more inelastic collision.

5. Can the initial velocities of the balls affect the outcome of a collision?

Yes, the initial velocities of the balls can affect the outcome of a collision. The higher the initial velocities, the more energy will be transferred during the collision. This can result in a more elastic or inelastic collision, depending on the coefficient of restitution and the mass and direction of motion of the balls.

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