Collisions between two balls in 2d-motion

[zeralda21]

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?

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...

 

[DrClaude]

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.

 

[zeralda21]

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;

$$v_0cosα=1$$
$$1-1/2gt^2+v_0tsinα=2-1/2gt^2 ⇔ v_0tsinα=1$$

Thus we end up with the system of equations: $$\left\{\begin{matrix}<br /> v_0\cos\alpha=1 & \\ <br /> v_0t\sin\alpha=1 & <br /> \end{matrix}\right.$$ which clearly does not have solutions.

 

[DrClaude]

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

$$v_0cosα=1$$

You're missing a $t$ in there, from which you can get $t_c$, the time at collision.

 

[zeralda21]

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;
$$\left\{\begin{matrix}<br /> v_0t\cos\alpha=1 & \\ <br /> v_0t\sin\alpha=1 & <br /> \end{matrix}\right.$$ leads to $$\tan\alpha=1 \Rightarrow \alpha=\frac{\pi}{4}$$

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.

 

[DrClaude]

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.

 

[consciousness]

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!

 

[zeralda21]

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.

 

[consciousness]

Nice! I like this approach better.

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