Intro to Relative Velocity: Videos & Practice Problems

Relative motion involves calculating the velocity of one object concerning another, using simple addition or subtraction of velocities. The relative velocity equation, $v\_ac=v\_a\_b+v\_b\_c$, helps in solving these problems. Understanding the frame of reference is crucial, as it determines how velocities are measured, leading to insights into concepts like acceleration and momentum in various contexts.

Intro to Relative Motion (Relative Velocity)

Understanding relative motion is essential for calculating the velocity of an object in relation to another. This concept revolves around the idea that velocity is always measured with respect to a reference point, known as a frame of reference. Typically, this frame of reference is the ground or Earth unless specified otherwise.

To illustrate relative motion, consider an observer on a moving walkway, such as those found in airports, measuring the velocities of three individuals: one standing still, one walking to the right, and one walking to the left. At time \( t = 0 \), all individuals are aligned, but after one second, the walkway moves them to different positions based on their actions.

For the stationary person (Person A), the walkway moves them at a velocity of 3 meters per second. Since they are not moving relative to the walkway, their velocity relative to the observer is also 3 meters per second. This demonstrates that when a stationary object is on a moving platform, they share the same velocity, resulting in a relative velocity of zero between them.

Now, consider Person B, who walks forward at 2 meters per second. In the same time frame, they cover more distance due to their walking speed. Therefore, when measured by the observer, their total velocity is the sum of their walking speed and the walkway's speed: \( 3 \, \text{m/s} + 2 \, \text{m/s} = 5 \, \text{m/s} \).

Conversely, Person C walks in the opposite direction at 2 meters per second. Their movement reduces the effective velocity relative to the walkway. Thus, when measured, their velocity is calculated as \( 3 \, \text{m/s} - 2 \, \text{m/s} = 1 \, \text{m/s} \). This negative contribution reflects the opposing direction of their movement.

In summary, relative velocity is determined by the simple addition or subtraction of velocities, depending on their directions. This foundational concept is crucial for solving problems involving motion and understanding how different frames of reference affect velocity measurements.

Solving Relative Velocity Problems in 1D

Understanding relative motion is crucial for solving problems involving different reference frames. The concept of relative velocity allows us to analyze how the speed of one object appears from the perspective of another. To effectively tackle relative velocity problems in one dimension, we utilize the relative velocity equation, which can be expressed as:

$$ v_{ac} = v_{ab} + v_{bc} $$

In this equation, \( v_{ac} \) represents the velocity of object A relative to object C, \( v_{ab} \) is the velocity of object A relative to object B, and \( v_{bc} \) is the velocity of object B relative to object C. While different texts may use varying symbols, the underlying principles remain consistent. It is essential to focus on the rules for setting up the equation rather than memorizing specific letters.

The first rule states that the inner subscripts of the terms on the right side of the equation must be the same. For example, if we have \( v_{ab} \), both subscripts 'b' must match. The second rule indicates that the outer subscripts on the right side must correspond to the subscripts on the left side. Thus, if we are looking for \( v_{ac} \), the outer subscripts must be 'a' and 'c'. Following these rules ensures the correct setup of the equation.

To illustrate this, consider a scenario where a car is moving at 45 meters per second east relative to the ground, while a truck ahead is traveling at 60 meters per second in the same direction. To find the velocity of the truck relative to the car, we first identify the objects and their velocities. The car's velocity relative to the ground is denoted as \( v_{cg} = 45 \, \text{m/s} \) and the truck's velocity as \( v_{tg} = 60 \, \text{m/s} \).

Next, we need to express the target variable, which is the velocity of the truck relative to the car, written as \( v_{tc} \). Using the relative velocity equation, we can set it up as:

$$ v_{tc} = v_{tg} + v_{gc} $$

Here, \( v_{gc} \) is the velocity of the ground relative to the car. Since we have \( v_{cg} \), we can find \( v_{gc} \) by reversing the subscripts, which changes the sign:

$$ v_{gc} = -v_{cg} = -45 \, \text{m/s} $$

Now, substituting the known values into the equation gives us:

$$ v_{tc} = 60 \, \text{m/s} + (-45 \, \text{m/s}) = 15 \, \text{m/s} $$

This result indicates that the truck is moving 15 meters per second faster than the car from the car's perspective. Thus, while the truck's speed relative to the ground is 60 meters per second, the car measures the truck's speed as 15 meters per second ahead of it.

In summary, mastering the setup of the relative velocity equation and understanding the relationships between different reference frames is key to solving relative motion problems effectively.

A boat on a river is traveling from a pier to a point 500 m upstream (against the river's current). The current flows at 4 m/s. If the boat makes the trip in 250 s, what is the speed of the boat relative to the water?

• A. 2 m/s
• B. 0 m/s
• C. 6 m/s
• D. 12 m/s

Answer: (C)

Flying in the Wind

In this scenario, we analyze the impact of wind on the travel time of an airliner between two cities, A and B, which are 2775 kilometers apart. Initially, without wind, the airliner takes 3.3 hours to complete the journey. To find the airliner's speed in still air, we calculate it using the formula for velocity:

$$ V_{PA} = \frac{\Delta x}{\Delta t} = \frac{2775 \text{ km}}{3.3 \text{ hours}} \approx 841 \text{ km/h} $$

When a wind of 225 km/h blows from west to east, we need to determine the new travel time, denoted as \( \Delta t_{final} \). The wind affects the airliner's effective speed relative to the ground, which we denote as \( V_{PG} \). The relationship between the velocities can be expressed as:

$$ V_{PG} = V_{PA} + V_{AG} $$

Here, \( V_{AG} \) is the velocity of the air (wind) relative to the ground. Substituting the known values:

$$ V_{PG} = 841 \text{ km/h} + 225 \text{ km/h} = 1066 \text{ km/h} $$

Now, we can find the new travel time using the distance and the new velocity:

$$ \Delta t_{final} = \frac{\Delta x}{V_{PG}} = \frac{2775 \text{ km}}{1066 \text{ km/h}} \approx 2.6 \text{ hours} $$

This result indicates that the wind assists the airliner, reducing the travel time from 3.3 hours to approximately 2.6 hours. Thus, the presence of wind significantly enhances the efficiency of the flight, demonstrating the importance of considering relative velocities in motion problems.

Here’s what students ask on this topic:

What is relative velocity and how is it calculated?

Relative velocity is the velocity of one object as observed from another moving object. It is calculated by adding or subtracting the velocities of the objects involved, depending on their directions. The general formula for relative velocity is given by:

$v_{\mathrm{ac}}=v_{\mathrm{ab}}+v_{\mathrm{bc}}$

Here, $v_{\mathrm{ac}}$ is the velocity of object A relative to object C, $v_{\mathrm{ab}}$ is the velocity of object A relative to object B, and $v_{\mathrm{bc}}$ is the velocity of object B relative to object C. Understanding the frame of reference is crucial for accurate calculations.

How do you solve relative velocity problems in one dimension?

To solve relative velocity problems in one dimension, follow these steps:

1. Draw a diagram and identify all objects and reference frames.
2. Write each given velocity using subscript notation, indicating the object and its reference frame (e.g., $v_{\mathrm{cg}}$ for the velocity of the car relative to the ground).
3. Set up the relative velocity equation using the rules for inner and outer subscripts:

$v_{\mathrm{tc}}=v_{\mathrm{tg}}+v_{\mathrm{gc}}$

Here, $v_{\mathrm{tc}}$ is the velocity of the truck relative to the car, $v_{\mathrm{tg}}$ is the velocity of the truck relative to the ground, and $v_{\mathrm{gc}}$ is the velocity of the ground relative to the car.

If needed, reverse the subscripts and change the sign of the velocity. Finally, solve the equation to find the desired relative velocity.

What is a frame of reference in relative velocity problems?

A frame of reference is a coordinate system or viewpoint from which the position and motion of objects are measured. In relative velocity problems, the frame of reference determines how velocities are observed and calculated. For example, if you are standing on the ground and measuring the speed of a car, the ground is your frame of reference. Most problems use the ground or Earth as the default frame of reference unless stated otherwise. Understanding the frame of reference is crucial for accurately determining relative velocities and solving related problems.

How do you handle opposite directions in relative velocity calculations?

When dealing with opposite directions in relative velocity calculations, you need to assign a negative sign to the velocity of the object moving in the opposite direction. For example, if one object is moving at 3 m/s to the right and another at 2 m/s to the left, their relative velocity is calculated by subtracting the leftward velocity from the rightward velocity:

$3-2=1$ m/s.

This approach ensures that the directions are correctly accounted for, leading to accurate relative velocity calculations.

What is the significance of the relative velocity equation?

The relative velocity equation is significant because it provides a systematic way to calculate the velocity of one object relative to another. The equation:

$v_{\mathrm{ac}}=v_{\mathrm{ab}}+v_{\mathrm{bc}}$

allows you to break down complex motion into simpler components, making it easier to solve problems involving multiple moving objects. By understanding and applying this equation, you can accurately determine how objects move relative to each other, which is essential for analyzing motion in various contexts, such as transportation, sports, and physics experiments.