# Analysis of Billiards Ball Motion Following a Horizontal Queue Impact

• besnik93
In summary, the conversation discusses a cue hitting a billiards ball with a force F, causing a shock that lasts for a very short time Δt. The ball, with radius R and mass M, has a moment of inertia I = 2/5 * M * R^2 with respect to its center of mass. After the shock, the ball has a combination of translational and rotational movement. The problem then asks to determine the speed of the ball's center of mass and its angular momentum with respect to the center of mass after the collision, as well as the height at which the cue should hit the ball so that it immediately rolls without slipping. The solution involves the conservation of linear and angular momentum, with the use of
besnik93
1. Encountered with cue to a massive billiards ball, which is initially at rest, see figure uploaded. The ball has radius R and mass M. The queue hits with force F horizontally into the bale height h above the table, and the shock lasts a very short time Δt.
It is reported that the moment of inertia of the ball with respect to its center of mass is
I = 2/5 * M * R^2

The movement after the shock is a combination of a translational movement and a rotation about an axis through the center of gravity perpendicular to the plane of the paper.

a) Determine the speed of the billiard ball's center of mass and billiard ball's angular momentum with respect to the center of mass immediately after the collision. The answers must be expressed by the known sizes M, h, R, F and Δt.

b) At what height should the queue hit the ball to the ball immediately after the collision rolls without slipping?

## The Attempt at a Solution

a) i think of focusing on the the center of mass, but how, i don't know..

b) i know that i need to focus on the expression of the mass center point of the speed and bale angular velocity. But i can't move on.

so i hope someone can help me, please..

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Note that the change in linear momentum = F x Δt, called an impulse. The change in angular momentum would equal T x Δt, where T equals torque. Can you express torque as an equation using F, h, and R?

I don't know how to express that to make sense

You guessed right with using the center of mass as the reference point.

The speed of the ball after the perfectly elastic collision is a very simple conservation of momentum:

F Δt = M v

The speed of the ball's angular momentum would then be conservation of angular momentum where you take the moment or torque about the ball's center of mass:

F Δt (h-R) = I ω

I think question b) is not stated quite correctly. I think it should read as follows:

b) At what height should the queue hit the ball so that the ball immediately after the collision rolls without slipping?

b) yes that's it

## 1. How does the horizontal queue impact affect the motion of billiards balls?

The horizontal queue impact, or the strike of the cue stick on the horizontal axis of the ball, affects the motion of billiards balls by transferring energy from the cue stick to the ball. This energy causes the ball to move in a specific direction and with a certain velocity.

## 2. What factors can influence the resulting motion of billiards balls after a horizontal queue impact?

The resulting motion of billiards balls after a horizontal queue impact can be influenced by several factors, including the angle of impact, the force of the cue stick, the surface of the table, and the friction between the cue ball and the object ball.

## 3. How is the trajectory of the billiards ball affected by a horizontal queue impact?

The trajectory of the billiards ball is affected by a horizontal queue impact in two main ways. First, the angle of impact determines the initial direction of the ball's motion. Second, the force of the cue stick and the friction between the balls and the table affect the speed and distance the ball travels.

## 4. What mathematical equations are used to analyze the motion of billiards balls after a horizontal queue impact?

The motion of billiards balls after a horizontal queue impact is typically analyzed using the laws of motion, specifically Newton's second law of motion (F=ma) and the law of conservation of momentum (p=mv). Additionally, equations for projectile motion and friction can also be used to determine the trajectory and speed of the balls.

## 5. How can the analysis of billiards ball motion following a horizontal queue impact be applied in real-world situations?

The analysis of billiards ball motion following a horizontal queue impact can be applied in real-world situations such as pool tournaments, where players need to accurately predict the trajectory and speed of the balls in order to make successful shots. This analysis can also be useful in other sports or activities that involve the movement of objects, such as bowling, baseball, or even car collisions.

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