Question

An ant walks due east at a constant speed of \(2 \mathrm{mi} / \mathrm{hr}\) on a sheet of paper that rests on a table. Suddenly, the sheet of paper starts moving southeast at \(\sqrt{2} \mathrm{mi} / \mathrm{hr} .\) Describe the motion of the ant relative to the table.

Short answer

Answer: The ant moves at a speed of \(\sqrt{2} \mathrm{mi} / \mathrm{hr}\) at a 45-degree angle (northeast) relative to the table.

Step-by-step solution

Identifying velocities

The given velocities are: - Velocity of ant with respect to paper: \(2 \mathrm{mi} / \mathrm{hr}\) (due east) - Velocity of paper with respect to table: \(\sqrt{2} \mathrm{mi} / \mathrm{hr}\) (southeast)

Converting velocities to components

Due East has an angle of 0 degrees and Southeast has an angle of 135 degrees. We can use the following formulas to find the components: \(V_x = V * \cos{\theta}\) \(V_y = V * \sin{\theta}\) Components for the velocity of ant with respect to paper: \(V_{ax} = 2 * \cos{0} = 2 \mathrm{mi} / \mathrm{hr}\) \(V_{ay} = 2 * \sin{0} = 0 \mathrm{mi} / \mathrm{hr}\) Components for the velocity of paper with respect to table: \(V_{px} = \sqrt{2} * \cos{135} = -1 \mathrm{mi} / \mathrm{hr}\) \(V_{py} = \sqrt{2} * \sin{135} = 1 \mathrm{mi} / \mathrm{hr}\)

Calculating the ant's velocity relative to the table

Add the components of the two velocities: \(V_{atx} = V_{ax} + V_{px} = 2 - 1 = 1 \mathrm{mi} / \mathrm{hr}\) \(V_{aty} = V_{ay} + V_{py} = 0 + 1 = 1 \mathrm{mi} / \mathrm{hr}\) The ant's velocity relative to the table in component form is \(1 \mathrm{mi} / \mathrm{hr}\) (east) and \(1 \mathrm{mi} / \mathrm{hr}\) (north).

Calculating magnitude and direction

The magnitude of the ant's velocity relative to the table can be found using Pythagorean theorem: \(V_{at} = \sqrt{V_{atx}^2 + V_{aty}^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \mathrm{mi} / \mathrm{hr}\) The direction can be found using the formula: \(\theta = \arctan(\frac{V_{aty}}{V_{atx}}) = \arctan(\frac{1}{1}) = 45^\circ\) Thus, the ant moves at a speed of \(\sqrt{2} \mathrm{mi} / \mathrm{hr}\) at a 45-degree angle (northeast) relative to the table.

Key concepts

Velocity Components

In physics, velocity is a vector quantity, which means it has both a magnitude (speed) and a direction. To solve complex problems involving motion, it's often useful to break down a velocity vector into its horizontal (x-component) and vertical (y-component) parts. This process is known as finding the velocity components. For example, in our exercise involving an ant on a moving sheet of paper, the ant's velocity eastward (x-direction) is constant at 2 mi/hr, while the paper's southeastward motion has components in both the x and y directions (south and east).

Using trigonometric functions, we find the x and y components by multiplying the velocity by the cosine of the angle for the x-component, and by the sine of the angle for the y-component. By analyzing motion through its components, we can assess situations from a new perspective, simplifying problems that may seem complex at first glance.

Pythagorean Theorem in Velocity

Once we have the velocity components, we can combine them to find the result of two motions taking place simultaneously. In cases where the directions of movement are perpendicular to each other, the Pythagorean theorem comes in handy. This theorem, which states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides, applies not only to distances but also to velocity.

In our exercise, after finding the x and y velocity components of the ant relative to the table, we can calculate the ant's total velocity magnitude using the Pythagorean theorem: \( V_{at} = \sqrt{V_{atx}^2 + V_{aty}^2} \). This formula allows us to determine the ant's speed relative to the table in a straightforward manner. It provides a clear and concrete understanding of how separate movements combine to produce a resultant velocity.

Vector Addition of Velocities

Vector addition of velocities is the method of combining multiple velocity vectors to determine the total or resultant velocity. In practice, when an object is in motion and an external velocity is imparted, we add the velocity vectors to analyze the resulting motion relative to a frame of reference. This concept is critical to our exercise, as the ant's motion relative to the paper needs to be combined with the paper's motion relative to the table.

To add the velocities, their x and y components are summed up separately: \(V_{atx} = V_{ax} + V_{px}\) and \(V_{aty} = V_{ay} + V_{py}\). This step gives us the components of the ant's resultant velocity. Through vector addition, we gained a comprehensive picture of the overall motion. Understanding vector addition is essential in many fields, including navigation, where it is used to account for wind speed and direction when steering a course.