Pool ball hits table, find an angle

• mbrmbrg
In summary, Figure 9-47 shows a cue ball with an initial speed of 1.60 m/s and an angle of 30.0° bouncing off the rail of a pool table. The y component of the ball's velocity is reversed, but the x component remains the same. When asked for theta_2, the correct answer is 30 degrees. The change in the ball's linear momentum can be expressed in unit-vector notation. However, the notation used in the diagram may be unfamiliar to some and may require conversion to a more familiar coordinate system.
mbrmbrg
Figure 9-47 (attatched) shows a 0.165 kg cue ball as it bounces from the rail of a pool table. The ball's initial speed is 1.60 m/s, and the angle 1 = 30.0°. The bounce reverses the y component of the ball's velocity but does not alter the x component.

(a) What is theta_2?
(b) What is the change in the ball's linear momentum in unit-vector notation?

I got the physics of part b, the trig of part a is messing me up. I went so far as to make a system:
$$v\cos\theta_2=-v\cos\theta_1$$
$$v\sin\theta_2=v\sin\theta_1$$

As I thought! The angle must be in the second quadrant... so that means add 90 degrees, right? (And not 180-angle?)
But I don't know how to go from the typical coordinate system used for measuring angles to the funky one shown in the diagram.

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Never mind. I got the correct answer of 30 degrees (I had started out by saying theta_2 = -30 degrees). But why? How is that allowed? What kind of crazy notation are they using here (that apparently is standard...)?

I would approach this problem by breaking it down into smaller parts and using the principles of physics to find the answers.

(a) To find theta_2, we can use the law of reflection which states that the angle of incidence is equal to the angle of reflection. In this case, the angle of incidence is 30 degrees, so the angle of reflection, theta_2, would also be 30 degrees. Therefore, theta_2 = 30 degrees.

(b) To find the change in the ball's linear momentum, we can use the equation p = mv, where p is the linear momentum, m is the mass of the ball, and v is the velocity. In this case, the mass of the ball is 0.165 kg and the initial velocity is 1.60 m/s. Since the x component of the velocity remains unchanged, the change in the ball's linear momentum in the x direction would be 0. In the y direction, the initial velocity is in the positive direction, but after the bounce it is in the negative direction. This means that there is a change in the y component of the ball's velocity. Using the law of reflection again, we can see that the y component of the velocity is reversed, so the change in the y component would be -2v_0sin(theta_2), where v_0 is the initial velocity. In unit-vector notation, the change in the ball's linear momentum would be (0, -2v_0sin(theta_2)).

In terms of the funky coordinate system shown in the diagram, we can see that the x and y components are flipped. This means that the change in the ball's linear momentum in this coordinate system would be (-2v_0sin(theta_2), 0).

In conclusion, to solve this problem, we used the principles of physics such as the law of reflection and the equation for linear momentum to find the answers. It is important to approach problems like this systematically and use known principles to guide our thinking.

What is the angle of reflection when a pool ball hits a table?

The angle of reflection when a pool ball hits a table is equal to the angle of incidence, which is the angle at which the ball strikes the table surface.

How do you calculate the angle of reflection?

The angle of reflection can be calculated using the law of reflection, which states that the angle of incidence is equal to the angle of reflection. This can be represented by the equation: θr = θi, where θr is the angle of reflection and θi is the angle of incidence.

Is the angle of reflection affected by the speed of the pool ball?

No, the angle of reflection is not affected by the speed of the pool ball. The law of reflection only depends on the angle of incidence and the surface that the ball is hitting.

What factors can affect the angle of reflection of a pool ball?

The angle of reflection can be affected by the surface that the pool ball is hitting, as well as any imperfections or irregularities on the surface, such as bumps or dents. The angle of incidence can also be affected by the spin or English put on the ball by the player.

How does the angle of reflection affect the path of the pool ball?

The angle of reflection plays a crucial role in determining the path of the pool ball after it hits the table. The ball will bounce off the table at the same angle that it strikes the surface, resulting in a predictable trajectory for the ball.

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