Relative Velocity

Welcome, students! 🚀 Today, we’re diving into the fascinating world of relative velocity. This lesson will help you understand how motion looks different depending on the observer’s frame of reference. By the end, you’ll be able to calculate relative velocities and apply these concepts to real-world scenarios like moving trains, airplanes, and even river crossings. Ready to shift your perspective? Let’s go!

Understanding Frames of Reference

Before we jump into equations, let’s get familiar with the idea of a frame of reference. Imagine you’re sitting on a train, looking out the window. You see a tree standing still outside. From your perspective, the tree appears to be moving backward as the train speeds forward. But from someone standing on the ground, that tree is completely still.

This difference in how motion is perceived is due to the frame of reference. A frame of reference is simply a point of view from which you measure motion. There are two key types:

1. Inertial Frame of Reference: This is a frame that is either at rest or moving at a constant velocity. For example, a person standing still on the ground or a train moving at a constant speed in a straight line.

1. Non-Inertial Frame of Reference: This is a frame that is accelerating. For example, if the train suddenly speeds up or slows down, or if it’s going around a curve, it’s a non-inertial frame.

Let’s explore how motion changes depending on the frame of reference.

Real-World Example: The Bus Ride

Imagine you’re sitting on a bus traveling at 20 m/s. You toss a ball straight up into the air. From your perspective (inside the bus), the ball goes up and comes straight back down. But from someone standing on the sidewalk, the ball follows a curved path because it’s moving forward with the bus as well as up and down.

This is the essence of relative velocity: the velocity of an object depends on the velocity of the observer.

The Concept of Relative Velocity

Relative velocity is the velocity of one object as observed from another moving object. It’s all about comparing velocities from different frames of reference.

Let’s break it down with a simple formula. If you have two objects, A and B, with velocities $v_A$ and $v_B$ (both measured relative to the ground), the relative velocity of A with respect to B is:

$$

$v_{AB} = v_A - v_B$

$$

This equation tells us how fast A appears to be moving when you’re riding along with B.

Example: Two Cars on a Highway

Let’s say Car A is driving east at 30 m/s, and Car B is driving east at 20 m/s. From the ground, their velocities are:

• $v_A = 30 \, \text{m/s}$
• $v_B = 20 \, \text{m/s}$

The relative velocity of Car A with respect to Car B is:

$$

v_{AB} = v_A - v_B = 30 \, $\text{m/s}$ - 20 \, $\text{m/s}$ = 10 \, $\text{m/s}$

$$

So, from the perspective of Car B’s driver, Car A appears to be moving forward at 10 m/s.

Opposite Directions

Now, let’s change the scenario. Suppose Car A is still driving east at 30 m/s, but Car B is driving west at 20 m/s. What’s the relative velocity now?

$$

v_{AB} = v_A - v_B = 30 \, $\text{m/s}$ - (-20 \, $\text{m/s}$) = 30 \, $\text{m/s}$ + 20 \, $\text{m/s}$ = 50 \, $\text{m/s}$

$$

In opposite directions, the relative velocity is larger. From Car B’s perspective, Car A is zooming past at 50 m/s!

Relative Velocity in Two Dimensions

So far, we’ve looked at motion in one dimension (along a straight line). But what if things are moving in two dimensions, like a plane flying in the wind or a boat crossing a river?

Velocity as a Vector

Velocity is a vector, meaning it has both magnitude (speed) and direction. When dealing with two dimensions, we need to consider both components: the $x$-component (horizontal) and the $y$-component (vertical).

We can represent a velocity vector as:

$$

$\mathbf{v} = (v_x, v_y)$

$$

Where $v_x$ is the velocity in the horizontal direction and $v_y$ is the velocity in the vertical direction.

Example: The Airplane and the Wind

Suppose an airplane is flying north at 100 m/s. There’s a wind blowing east at 30 m/s. What’s the plane’s velocity relative to the ground?

We’ll combine the two velocity vectors. Let’s call the plane’s velocity $\mathbf{v_p} = (0, 100) \, \text{m/s}$ (since it’s moving north, there’s no east/west component). The wind’s velocity is $\mathbf{v_w} = (30, 0) \, \text{m/s}$.

The resultant velocity (plane’s velocity relative to the ground) is:

$$

$\mathbf{v_{pg}}$ = $\mathbf{v_p}$ + $\mathbf{v_w}$ = (0 + 30, 100 + 0) = (30, 100) \, $\text{m/s}$

$$

Now, to find the magnitude of this resultant velocity, we use the Pythagorean theorem:

$$

|$\mathbf{v_{pg}}$| = $\sqrt{(30)^2 + (100)^2}$ = $\sqrt{900 + 10000}$ = $\sqrt{10900}$ $\approx 104$.4 \, $\text{m/s}$

$$

To find the direction, we use the inverse tangent function:

$$

$\theta$ = \tan^{-1}$\left($$\frac{v_y}{v_x}$$\right)$ = \tan^{-1}$\left($$\frac{100}{30}$$\right)$ $\approx 73$.3^$\circ$

$$

So, the plane is flying at about 104.4 m/s at an angle of 73.3° north of east.

Example: The Boat Crossing the River

Let’s consider another classic example: a boat crossing a river. Suppose the boat can move at 5 m/s relative to the water, and the river is flowing east at 3 m/s.

We want to find the boat’s velocity relative to the shore. Let’s assume the boat is heading straight north (perpendicular to the river current). The boat’s velocity vector is $\mathbf{v_b} = (0, 5) \, \text{m/s}$, and the river’s velocity is $\mathbf{v_r} = (3, 0) \, \text{m/s}$.

The resultant velocity is:

$$

$\mathbf{v_{bs}}$ = $\mathbf{v_b}$ + $\mathbf{v_r}$ = (0 + 3, 5 + 0) = (3, 5) \, $\text{m/s}$

$$

The magnitude of the resultant velocity is:

$$

|$\mathbf{v_{bs}}$| = $\sqrt{(3)^2 + (5)^2}$ = $\sqrt{9 + 25}$ = $\sqrt{34}$ $\approx 5$.83 \, $\text{m/s}$

$$

The direction is:

$$

$\theta = \tan^{-1}\left(\frac{5}{3}\right) \approx 59.0^\circ$

$$

So, the boat’s velocity relative to the shore is about 5.83 m/s at an angle of 59.0° north of east. This means the boat will drift downstream while it’s crossing the river.

Adjusting for Drift

What if we want the boat to go straight across the river, with no downstream drift? We need to adjust the boat’s heading.

We want the resultant velocity in the east direction to be zero. So, we need to find the angle at which the boat should head upstream (west of north) to cancel out the river’s flow.

Let $\theta$ be the angle the boat must head upstream. We know that the eastward component of the boat’s velocity must cancel the river’s flow. That means:

$$

$5 \sin(\theta) = 3$

$$

Solving for $\theta$:

$$

$\sin(\theta) = \frac{3}{5} = 0.6$

$$

$$

$\theta = \sin^{-1}(0.6) \approx 36.9^\circ$

$$

So, the boat should head about 36.9° west of north to go straight across the river.

Relative Velocity in Daily Life

Relative velocity isn’t just a physics classroom concept—it’s all around us. Here are some real-world examples that you’ve probably encountered:

Walking on a Moving Walkway

Imagine you’re at an airport, walking on a moving walkway. Let’s say the walkway moves at 1 m/s, and you walk forward at 1.5 m/s relative to the walkway. What’s your velocity relative to the ground?

$$

v_{relative} = v_{walkway} + v_{you} = 1 \, $\text{m/s}$ + 1.5 \, $\text{m/s}$ = 2.5 \, $\text{m/s}$

$$

So, you’re moving at 2.5 m/s relative to the ground. But if you turn around and walk against the walkway at 1.5 m/s:

$$

v_{relative} = v_{walkway} - v_{you} = 1 \, $\text{m/s}$ - 1.5 \, $\text{m/s}$ = -0.5 \, $\text{m/s}$

$$

Now you’re moving backward relative to the ground at 0.5 m/s!

Escalators

If you’ve ever tried to walk up a down escalator, you’ve experienced relative velocity. Let’s say the escalator is going down at 2 m/s, and you’re walking up at 2 m/s. What’s your velocity relative to the ground?

$$

v_{relative} = v_{escalator} + v_{you} = -2 \, $\text{m/s}$ + 2 \, $\text{m/s}$ = 0 \, $\text{m/s}$

$$

You’re not moving relative to the ground at all! You’re essentially staying in the same place.

Sports: A Tennis Ball and a Racket

In sports, relative velocity plays a huge role. Imagine a tennis player hitting a serve. The racket is moving forward at 30 m/s, and the ball is coming toward the racket at 20 m/s. What’s the relative velocity of the ball with respect to the racket?

$$

v_{relative} = v_{racket} - (-v_{ball}) = 30 \, $\text{m/s}$ + 20 \, $\text{m/s}$ = 50 \, $\text{m/s}$

$$

The ball’s velocity relative to the racket is 50 m/s, which increases the impact force and sends the ball flying!

Conclusion

We’ve explored the exciting world of relative velocity, students! We started by understanding frames of reference and how motion looks different depending on your point of view. Then, we dove into the key equation $v_{AB} = v_A - v_B$ and applied it to real-world examples like cars on highways and airplanes in the wind.

We also expanded into two dimensions, using vectors to solve problems involving boats crossing rivers and planes flying in windy conditions. Finally, we saw how relative velocity shows up in everyday life, from moving walkways to sports.

By mastering these concepts, you’ll be able to analyze motion in all sorts of situations—and maybe even impress your friends with your new physics knowledge! 🌟

Study Notes

• Frame of Reference: The perspective from which motion is observed (e.g., ground, moving train).
• Relative Velocity Formula:

$$

$ v_{AB} = v_A - v_B$

$$

This shows the velocity of object A relative to object B.

• Same Direction:

$$

$ v_{AB} = v_A - v_B$

$$

If two objects move in the same direction, subtract their velocities.

• Opposite Direction:

$$

v_{AB} = v_A - (-v_B) = v_A + v_B

$$

If two objects move in opposite directions, add their velocities.

• Vector Addition in 2D:

For velocities in two dimensions, add components separately:

$$

\mathbf{v_{resultant}} = (v_{x1} + v_{x2}, v_{y1} + v_{y2})

$$

• Magnitude of a Velocity Vector:

$$

$ |\mathbf{v}| = \sqrt{v_x^2 + v_y^2}$

$$

• Direction of a Velocity Vector:

$$

$ \theta = \tan^{-1}\left(\frac{v_y}{v_x}\right)$

$$

• Airplane and Wind Example:

If a plane’s velocity is $\mathbf{v_p} = (0, 100) \, \text{m/s}$ north and wind’s velocity is $\mathbf{v_w} = (30, 0) \, \text{m/s}$ east, the resultant velocity is:

$$

$ \mathbf{v_{pg}} = (30, 100) \, \text{m/s}$

$$

Magnitude:

$$

$ |\mathbf{v_{pg}}| \approx 104.4 \, \text{m/s}$

$$

Direction:

$$

$\theta$ $\approx 73$.3^$\circ$ \, \text{north of east}

$$

• Boat Crossing River Example:

Boat’s velocity relative to water: $\mathbf{v_b} = (0, 5) \, \text{m/s}$, river’s velocity: $\mathbf{v_r} = (3, 0) \, \text{m/s}$.

Resultant velocity:

$$

$ \mathbf{v_{bs}} = (3, 5) \, \text{m/s}$

$$

Magnitude:

$$

$ |\mathbf{v_{bs}}| \approx 5.83 \, \text{m/s}$

$$

Direction:

$$

$\theta$ $\approx 59$.0^$\circ$ \, \text{north of east}

$$

• Adjusting Boat Heading:

To go straight across, find the angle $\theta$:

$$

$5 \sin($$\theta)$ = 3 \quad \Rightarrow \quad $\theta$ = $\sin^{-1}$(0.6) $\approx 36$.9^$\circ$

$$

• Relative Velocity in Everyday Life:
• Walking on a moving walkway: Add or subtract velocities depending on direction.
• Tennis ball and racket: Add the velocities of the ball and racket to find the impact speed.

With these formulas and concepts, you’re ready to tackle any relative velocity problem that comes your way! 🚀