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PhysicsPhysics151 views·Updated May 28, 2026·5 pages

2D Kinematics Notes - Chapter 3 from Physics with Algebra

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Rachael Murphy@writcha

Motion in multiple dimensions brings physics to life as we... Show more

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of 5
3.1 Displacement $ \Delta \vec{r} = \vec{r_2} - \vec{r_1} \longrightarrow \text{displacement vector} $ $\Delta \vec{r} = (x_2 - x_1)\hat{i}

Displacement and Velocity in 3D

Displacement in three dimensions is represented by a vector that points from your starting position to your final position. Mathematically, it's written as Δr⃗=r⃗2−r⃗1\vec{\Delta r} = \vec{r}_2 - \vec{r}_1 or broken down into components: Δr⃗=(x2−x1)i^+(y2−y1)j^+(z2−z1)k^\vec{\Delta r} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k}.

When studying motion, we use two types of velocity. Average velocity is calculated as displacement divided by time: v⃗avg=Δr⃗Δt\vec{v}_{avg} = \frac{\vec{\Delta r}}{\Delta t}. Instantaneous velocity gives us the velocity at any exact moment and is expressed as v⃗=vxi^+vyj^+vzk^\vec{v} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}.

Similarly, acceleration measures the rate of velocity change. Average acceleration is a⃗avg=Δv⃗Δt\vec{a}_{avg} = \frac{\vec{\Delta v}}{\Delta t}, while instantaneous acceleration is the derivative of velocity with respect to time: a⃗=dvxdti^+dvydtj^+dvzdtk^\vec{a} = \frac{dv_x}{dt}\hat{i} + \frac{dv_y}{dt}\hat{j} + \frac{dv_z}{dt}\hat{k}.

💡 The beauty of 3D motion is that we can analyze each dimension separately! The x, y, and z components of motion are independent of each other, making complex problems much more manageable.

2
of 5
3.1 Displacement $ \Delta \vec{r} = \vec{r_2} - \vec{r_1} \longrightarrow \text{displacement vector} $ $\Delta \vec{r} = (x_2 - x_1)\hat{i}

Problem-Solving Strategy and Projectile Motion

When tackling motion problems, follow this powerful strategy: make a drawing, establish positive/negative directions, write down known values for each dimension, verify you have at least three kinematic variables, and select the appropriate equation. Remember that when motion has segments, the final velocity of one segment becomes the initial velocity for the next.

Projectile motion occurs when an object moves through a vertical plane with an initial velocity while experiencing constant free-fall acceleration. This motion combines horizontal movement (with constant velocity) and vertical movement (with acceleration due to gravity).

When an object is launched with initial velocity v0v_0 at angle θ\theta, we can break this into components: vox=v0cos⁡θv_{ox} = v_0 \cos\theta and voy=v0sin⁡θv_{oy} = v_0 \sin\theta. The key insight is that horizontal and vertical motions are completely independent of each other.

🚀 Think of projectile motion like this: horizontally, the object moves as if there's no gravity, while vertically, it falls as if it were dropped from rest. This mental model makes visualizing complex trajectories much easier!

3
of 5
3.1 Displacement $ \Delta \vec{r} = \vec{r_2} - \vec{r_1} \longrightarrow \text{displacement vector} $ $\Delta \vec{r} = (x_2 - x_1)\hat{i}

Solving Projectile Motion Problems

The genius of projectile motion analysis is that we can convert a 2D problem into two separate 1D problems. For horizontal motion, acceleration is zero (ax=0a_x = 0), so velocity remains constant. For vertical motion, acceleration equals gravity (ay=−9.80m/s2a_y = -9.80 m/s^2).

To track a projectile's position, we use the standard kinematic equations for each direction. Horizontally: x−x0=voxtx-x_0 = v_{ox}t or x−x0=(v0cos⁡θ0)tx-x_0 = (v_0\cos\theta_0)t. Vertically: y−y0=voyt−12gt2y-y_0 = v_{oy}t - \frac{1}{2}gt^2 or y−y0=(v0sin⁡θ0)t−12gt2y-y_0 = (v_0\sin\theta_0)t - \frac{1}{2}gt^2.

The path of a projectile forms a parabola, which we can express as y=(tanθ0)x−gx22v02cos⁡2θ0y = (tan\theta_0)x - \frac{gx^2}{2v_0^2\cos^2\theta_0} by eliminating time from our equations. This equation lets us determine the projectile's position at any point during its flight.

🎯 When you need to find the maximum range (horizontal distance) of a projectile, use R=v02gsin⁡(2θ0)R = \frac{v_0^2}{g}\sin(2\theta_0). You'll discover that a launch angle of exactly 45° maximizes this range when launching and landing at the same height!

4
of 5
3.1 Displacement $ \Delta \vec{r} = \vec{r_2} - \vec{r_1} \longrightarrow \text{displacement vector} $ $\Delta \vec{r} = (x_2 - x_1)\hat{i}

Relative Motion

The way we measure position and velocity depends entirely on our frame of reference. When two observers (A and B) track an object (P), their measurements relate in specific ways that help us understand motion from different perspectives.

For motion along one dimension, the positions are related by xPA=xPB+xBAx_{PA} = x_{PB} + x_{BA}, where the subscripts indicate "as measured by." Taking the derivative of this equation gives us the relationship between velocities: vPA=vPB+vBAv_{PA} = v_{PB} + v_{BA}.

Interestingly, for non-accelerating reference frames, both observers will measure the same acceleration of the object: aPA=aPBa_{PA} = a_{PB}. This makes acceleration a more "universal" measurement than position or velocity.

🚗 Relative motion explains why a passenger in a car traveling at 60 mph sees a coffee cup on the dashboard as stationary, while someone standing on the roadside sees the same cup moving at 60 mph. Both observations are correct—they're just made from different reference frames!

5
of 5
3.1 Displacement $ \Delta \vec{r} = \vec{r_2} - \vec{r_1} \longrightarrow \text{displacement vector} $ $\Delta \vec{r} = (x_2 - x_1)\hat{i}

Relative Motion in 2D

Relative motion principles extend naturally to two dimensions by using vectors instead of scalar quantities. The relationship between positions in different frames becomes $\vec{r}{PA} = \vec{r}{PB} + \vec{r}_{BA}$.

Similarly, velocities in different reference frames relate through vector addition: $\vec{v}{PA} = \vec{v}{PB} + \vec{v}_{BA}$. This powerful equation helps us solve problems involving objects moving in different directions.

For non-accelerating reference frames, the acceleration equation simplifies to $\vec{a}{PA} = \vec{a}{PB}$, meaning observers in different frames will measure the same acceleration. This consistency makes acceleration particularly useful in physics analysis.

🌍 Imagine watching a friend play catch on a moving train. You (standing on the platform) and another passenger (sitting on the train) will see completely different ball trajectories, yet you'll both calculate the same acceleration due to gravity. This illustrates how reference frames fundamentally change our perception of motion!

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PhysicsPhysics151 views·Updated May 28, 2026·5 pages

2D Kinematics Notes - Chapter 3 from Physics with Algebra

user profile picture
Rachael Murphy@writcha

Motion in multiple dimensions brings physics to life as we explore how objects move through space over time. Understanding displacement, velocity, and acceleration in 3D allows us to analyze everything from a baseball's arc to a rocket's trajectory. These concepts... Show more

1
of 5
3.1 Displacement $ \Delta \vec{r} = \vec{r_2} - \vec{r_1} \longrightarrow \text{displacement vector} $ $\Delta \vec{r} = (x_2 - x_1)\hat{i}

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  • Access to all documents
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  • Join milions of students

Displacement and Velocity in 3D

Displacement in three dimensions is represented by a vector that points from your starting position to your final position. Mathematically, it's written as Δr⃗=r⃗2−r⃗1\vec{\Delta r} = \vec{r}_2 - \vec{r}_1 or broken down into components: Δr⃗=(x2−x1)i^+(y2−y1)j^+(z2−z1)k^\vec{\Delta r} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k}.

When studying motion, we use two types of velocity. Average velocity is calculated as displacement divided by time: v⃗avg=Δr⃗Δt\vec{v}_{avg} = \frac{\vec{\Delta r}}{\Delta t}. Instantaneous velocity gives us the velocity at any exact moment and is expressed as v⃗=vxi^+vyj^+vzk^\vec{v} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}.

Similarly, acceleration measures the rate of velocity change. Average acceleration is a⃗avg=Δv⃗Δt\vec{a}_{avg} = \frac{\vec{\Delta v}}{\Delta t}, while instantaneous acceleration is the derivative of velocity with respect to time: a⃗=dvxdti^+dvydtj^+dvzdtk^\vec{a} = \frac{dv_x}{dt}\hat{i} + \frac{dv_y}{dt}\hat{j} + \frac{dv_z}{dt}\hat{k}.

💡 The beauty of 3D motion is that we can analyze each dimension separately! The x, y, and z components of motion are independent of each other, making complex problems much more manageable.

2
of 5
3.1 Displacement $ \Delta \vec{r} = \vec{r_2} - \vec{r_1} \longrightarrow \text{displacement vector} $ $\Delta \vec{r} = (x_2 - x_1)\hat{i}

Sign up to see the content. It's free!

  • Access to all documents
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Problem-Solving Strategy and Projectile Motion

When tackling motion problems, follow this powerful strategy: make a drawing, establish positive/negative directions, write down known values for each dimension, verify you have at least three kinematic variables, and select the appropriate equation. Remember that when motion has segments, the final velocity of one segment becomes the initial velocity for the next.

Projectile motion occurs when an object moves through a vertical plane with an initial velocity while experiencing constant free-fall acceleration. This motion combines horizontal movement (with constant velocity) and vertical movement (with acceleration due to gravity).

When an object is launched with initial velocity v0v_0 at angle θ\theta, we can break this into components: vox=v0cos⁡θv_{ox} = v_0 \cos\theta and voy=v0sin⁡θv_{oy} = v_0 \sin\theta. The key insight is that horizontal and vertical motions are completely independent of each other.

🚀 Think of projectile motion like this: horizontally, the object moves as if there's no gravity, while vertically, it falls as if it were dropped from rest. This mental model makes visualizing complex trajectories much easier!

3
of 5
3.1 Displacement $ \Delta \vec{r} = \vec{r_2} - \vec{r_1} \longrightarrow \text{displacement vector} $ $\Delta \vec{r} = (x_2 - x_1)\hat{i}

Sign up to see the content. It's free!

  • Access to all documents
  • Improve your grades
  • Join milions of students

Solving Projectile Motion Problems

The genius of projectile motion analysis is that we can convert a 2D problem into two separate 1D problems. For horizontal motion, acceleration is zero (ax=0a_x = 0), so velocity remains constant. For vertical motion, acceleration equals gravity (ay=−9.80m/s2a_y = -9.80 m/s^2).

To track a projectile's position, we use the standard kinematic equations for each direction. Horizontally: x−x0=voxtx-x_0 = v_{ox}t or x−x0=(v0cos⁡θ0)tx-x_0 = (v_0\cos\theta_0)t. Vertically: y−y0=voyt−12gt2y-y_0 = v_{oy}t - \frac{1}{2}gt^2 or y−y0=(v0sin⁡θ0)t−12gt2y-y_0 = (v_0\sin\theta_0)t - \frac{1}{2}gt^2.

The path of a projectile forms a parabola, which we can express as y=(tanθ0)x−gx22v02cos⁡2θ0y = (tan\theta_0)x - \frac{gx^2}{2v_0^2\cos^2\theta_0} by eliminating time from our equations. This equation lets us determine the projectile's position at any point during its flight.

🎯 When you need to find the maximum range (horizontal distance) of a projectile, use R=v02gsin⁡(2θ0)R = \frac{v_0^2}{g}\sin(2\theta_0). You'll discover that a launch angle of exactly 45° maximizes this range when launching and landing at the same height!

4
of 5
3.1 Displacement $ \Delta \vec{r} = \vec{r_2} - \vec{r_1} \longrightarrow \text{displacement vector} $ $\Delta \vec{r} = (x_2 - x_1)\hat{i}

Sign up to see the content. It's free!

  • Access to all documents
  • Improve your grades
  • Join milions of students

Relative Motion

The way we measure position and velocity depends entirely on our frame of reference. When two observers (A and B) track an object (P), their measurements relate in specific ways that help us understand motion from different perspectives.

For motion along one dimension, the positions are related by xPA=xPB+xBAx_{PA} = x_{PB} + x_{BA}, where the subscripts indicate "as measured by." Taking the derivative of this equation gives us the relationship between velocities: vPA=vPB+vBAv_{PA} = v_{PB} + v_{BA}.

Interestingly, for non-accelerating reference frames, both observers will measure the same acceleration of the object: aPA=aPBa_{PA} = a_{PB}. This makes acceleration a more "universal" measurement than position or velocity.

🚗 Relative motion explains why a passenger in a car traveling at 60 mph sees a coffee cup on the dashboard as stationary, while someone standing on the roadside sees the same cup moving at 60 mph. Both observations are correct—they're just made from different reference frames!

5
of 5
3.1 Displacement $ \Delta \vec{r} = \vec{r_2} - \vec{r_1} \longrightarrow \text{displacement vector} $ $\Delta \vec{r} = (x_2 - x_1)\hat{i}

Sign up to see the content. It's free!

  • Access to all documents
  • Improve your grades
  • Join milions of students

Relative Motion in 2D

Relative motion principles extend naturally to two dimensions by using vectors instead of scalar quantities. The relationship between positions in different frames becomes $\vec{r}{PA} = \vec{r}{PB} + \vec{r}_{BA}$.

Similarly, velocities in different reference frames relate through vector addition: $\vec{v}{PA} = \vec{v}{PB} + \vec{v}_{BA}$. This powerful equation helps us solve problems involving objects moving in different directions.

For non-accelerating reference frames, the acceleration equation simplifies to $\vec{a}{PA} = \vec{a}{PB}$, meaning observers in different frames will measure the same acceleration. This consistency makes acceleration particularly useful in physics analysis.

🌍 Imagine watching a friend play catch on a moving train. You (standing on the platform) and another passenger (sitting on the train) will see completely different ball trajectories, yet you'll both calculate the same acceleration due to gravity. This illustrates how reference frames fundamentally change our perception of motion!

We thought you’d never ask...

What is the Knowunity AI companion?

Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Most popular content in Physics

9

Most popular content

9

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.6/5App Store
4.7/5Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan SiOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha KlichAndroid user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

AnnaiOS user