Question

A billiard ball of mass \(m=0.250 \mathrm{~kg}\) hits the cushion of a billiard table at an angle of \(\theta_{1}=60.0^{\circ}\) at a speed of \(v_{1}=27.0 \mathrm{~m} / \mathrm{s}\) It bounces off at an angle of \(\theta_{2}=71.0^{\circ}\) and a speed of \(v_{2}=10.0 \mathrm{~m} / \mathrm{s}\). a) What is the magnitude of the change in momentum of the billiard ball? b) In which direction does the change of momentum vector point?

Short answer

a) The magnitude of the change in momentum of the billiard ball is 4.315 kg.m/s. b) The direction of the change in momentum vector points at an angle of 54.1 degrees from the negative x-axis.

Step-by-step solution

Calculate the initial momentum components

Using the given information about the incoming angle and the speed of the billiard ball, we will find the components of the initial momentum (\(p_{1x}\) and \(p_{1y}\)) by using these equations: $$p_{1x} = m*v_{1}*cos(\theta_1)$$ $$p_{1y} = m*v_{1}*sin(\theta_1)$$ Plugging in the given values, $p_{1x} = 0.250 * 27.0 * cos(60^{\circ}) \\ = 0.250 * 27.0 * 0.5 \\ = 3.375 kg.m/s$ $p_{1y} = 0.250 * 27.0 * sin(60^{\circ}) \\ = 0.250 * 27.0 * \frac{\sqrt{3}}{2} \\ = 5.845 kg.m/s$

Calculate the final momentum components

Using the given information about the outgoing angle and speed of the billiard ball, we will find the components of the final momentum (\(p_{2x}\) and \(p_{2y}\)) by using these equations: $$p_{2x} = m*v_{2}*cos(\theta_2)$$ $$p_{2y} = m*v_{2}*sin(\theta_2)$$ Plugging in the given values, $p_{2x} = 0.250 * 10.0 * cos(71^{\circ}) \\ = 0.250 * 10.0 * 0.335 \\ = 0.837 kg.m/s$ $p_{2y} = 0.250 * 10.0 * sin(71^{\circ}) \\ = 0.250 * 10.0 * 0.942 \\ = 2.355 kg.m/s$

Calculate the change in momentum components

To find the change in momentum components, we will subtract the initial components from the final components: $$\Delta p_x = p_{2x} - p_{1x}$$ $$\Delta p_y = p_{2y} - p_{1y}$$ $\Delta p_x = 0.837 - 3.375 \\ = -2.538 kg.m/s$ $\Delta p_y = 2.355 - 5.845 \\ = -3.490 kg.m/s$

Calculate the magnitude of the change in momentum

We will use the Pythagorean theorem to find the magnitude of the change in momentum: $$|\Delta p| = \sqrt{(\Delta p_x)^2 + (\Delta p_y)^2}$$ $|\Delta p| = \sqrt{(-2.538)^2 + (-3.490)^2} \\ = \sqrt{6.433 + 12.180} \\ = \sqrt{18.613} \\ = 4.315 kg.m/s$ So, the magnitude of the change in momentum is 4.315 kg.m/s.

Calculate the direction of the change in momentum

We will find the angle of the change in momentum vector (\(\alpha\)) by using the arctangent function: $$\alpha = arctan\left(\frac{\Delta p_y}{\Delta p_x}\right)$$ $\alpha = arctan\left(\frac{-3.490}{-2.538}\right) \\ = arctan(1.375) \\ = 54.1^{\circ}$ So, the change in momentum vector points in a direction of \(54.1^{\circ}\) from the negative x-axis. To summarize: a) The magnitude of the change in momentum of the billiard ball is 4.315 kg.m/s. b) The change of momentum vector points in a direction of \(54.1^{\circ}\) from the negative x-axis.

Key concepts

Newton's Laws of Motion

Newton's Laws of Motion are fundamental principles that describe how objects move. They are core to understanding momentum and collisions, like in our billiard ball problem. There are three laws:
- **First Law (Law of Inertia):** An object will remain at rest or move at a constant velocity unless acted upon by an external force.
- **Second Law:** It states that the force acting on an object is equal to the mass of that object times its acceleration (\( F = ma \) ). This law relates to the momentum concept because momentum (\( p \) ) is the product of mass and velocity, \( p = mv \).
- **Third Law:** For every action, there is an equal and opposite reaction. This is crucial in understanding how forces interact, such as when a billiard ball hits a cushion, exerting equal and opposite forces.
In the context of this exercise, when the billiard ball bounces off the cushion, its change in velocity reflects the application of Newton's laws, resulting in a change in momentum. This is particularly explained by the second law, where the change in velocity leads to a change in momentum due to external forces acting on the ball.

Pythagorean Theorem

The Pythagorean Theorem is a mathematical sacred tool used to find the lengths of sides in a right triangle. It fundamentally states that in a right triangle, the square of the length of the hypotenuse (\( c \) ) is equal to the sum of the squares of the other two sides (\( a \) and \( b \) ):\[c^2 = a^2 + b^2\]
In our problem, this theorem helps us calculate the magnitude of the momentum change. After finding the individual momentum components (\( \Delta p_x \) and \( \Delta p_y \) ), we apply the Pythagorean theorem to get the resultant change in momentum vector:\[|\Delta p| = \sqrt{(\Delta p_x)^2 + (\Delta p_y)^2}\]
This formula tells us the magnitude of the overall change in momentum, compactly solving for \( |\Delta p| \). By doing so, it allows us to measure how the ball's motion is altered after bouncing off the table cushion.

Vector Components

Breaking forces, velocities, or momentum into their vector components allows us to handle vector quantities as they interact in multiple dimensions. Any vector component describes a vector in terms of horizontal (x-axis) and vertical (y-axis) parts.
Vectors are essential in accurately modeling the motion in physics problems. For instance, the momentum of a billiard ball moving in a plane is not a straightforward value; it involves distinct movements along different axes.
In the example, the initial and final momenta are each broken down into x and y components:

• Initial x-component: \( p_{1x} = m * v_1 * \cos(\theta_1) \)
• Initial y-component: \( p_{1y} = m * v_1 * \sin(\theta_1) \)
• Final x-component: \( p_{2x} = m * v_2 * \cos(\theta_2) \)
• Final y-component: \( p_{2y} = m * v_2 * \sin(\theta_2) \)

This breakdown is vital because it allows us to calculate the changes in these individual components, which leads to understanding the overall vector change and magnitude through further calculations and applications like the Pythagorean Theorem.

Trigonometry in Physics

Trigonometry helps us use angles in physics to solve problems where components act in different directions. Through sine, cosine, and tangent functions, we can separate a vector like velocity or force into its horizontal and vertical parts.
In our billiard ball problem, trigonometry helps determine how much of the ball's velocity or momentum acts in the x and y directions. The angle \( \theta_1 \) is utilized to calculate initial component vectors:

• \( p_{1x} = m*v_{1}*\cos(\theta_1) \)
• \( p_{1y} = m*v_{1}*\sin(\theta_1) \)

The outgoing angle \( \theta_2 \) gives us the final vector components.
Further, trigonometry is used to determine the direction of the vector change, understood through the inverse tangent (arctan) function:\[\alpha = \arctan\left(\frac{\Delta p_y}{\Delta p_x}\right)\]The approach enables us to handle complex directional changes in physical systems, making sure we account for all forces and movements accurately.