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Maths
2 Dec 2025
329
4 pages
Kinematics is all about tracking how objects move through space and time - something you experience every day... Show more
Think of kinematics like a GPS system for physics - it tracks position, velocity, and acceleration of moving objects. These three concepts are connected through differentiation and integration, making calculus your best mate for solving motion problems.
Displacement (r) tells you where a particle is located relative to a starting point. You can find this using coordinates (x, y) and calculate the distance with |r| = √. The displacement vector r is always underlined to show it's a vector quantity.
Velocity (v) shows how quickly position changes over time. You get this by differentiating displacement v = dr/dt = (ẋ, ẏ). The speed is just the magnitude |v| = √(ẋ² + ẏ²). Remember, velocity has direction whilst speed doesn't.
Acceleration (a) measures how velocity changes over time. It's the second derivative of displacement a = dv/dt = d²r/dt² = (ẍ, ÿ). This tells you if something's speeding up, slowing down, or changing direction.
Quick Tip Use dot notation (ẋ, ẏ) for first derivatives and double dots (ẍ, ÿ) for second derivatives - it's much faster than writing d/dt every time!
When two particles are moving through space, you can predict exactly when and where they'll cross paths by setting their position vectors equal to each other. It's like solving a collision problem in a video game!
For the example with particles E and F, you need to match up the i and j components separately. Set the i coefficients equal -11 + 5t = 13 + t, which gives you t = 6s. Then do the same for j components 23 - 3t = -7 + 2t, which also gives t = 6s.
Since both calculations give the same time, the particles definitely meet at t = 6 seconds. If you'd got different times, it would mean the particles never actually cross paths - they'd just pass by each other at different heights or distances.
To find the meeting position, substitute t = 6 back into either position vector. You should get the same answer from both particles m. This confirms your calculation is spot on!
Pro Tip Always check your answer by substituting the time into both position vectors - if they don't match, you've made an error somewhere!
Finding when particles are closest together or a specific distance apart involves working with relative displacement - basically tracking one particle from the perspective of the other. This technique is brilliant for collision avoidance problems!
Start by finding the relative displacement rₐ - rᵦ. This gives you a single vector equation that describes how far apart the particles are. In the example, this becomes i + j, which represents the separation between particles A and B.
The distance squared is d² = ² + ² = 2t² - 20t + 52. For part (a), set d² = 100 and solve the quadratic to get t = 2s and t = 12s. The particles are 10m apart at both these times!
For part (b), find when particles are closest by differentiating d² and setting it to zero d(d²)/dt = 4t - 20 = 0, giving t = 5s. At this moment, the particles are as close as they'll ever get during their motion.
Remember When finding minimum distance, you're looking for when the rate of change of distance squared equals zero - that's your closest approach!
Understanding motion in 2D means tracking both x and y coordinates as they change over time. Think of it like plotting a character's movement across a game map - you need both horizontal and vertical positions to know exactly where they are.
The position vector r = (x(t), y(t)) gives you coordinates at any time t. In the example, r = traces out a curved path as t goes from 0 to 4. You can literally plot this on a graph to see the particle's journey!
Velocity vectors act like tangent lines to the path, showing the direction of movement at each instant. When t = 3, v = (6, -1), and when t = 4, v = (8, -1). Notice how these vectors point along the curve's direction at those specific moments.
The acceleration vector a = (2, 0) stays constant throughout this motion, always pointing horizontally to the right. This constant rightward acceleration gradually curves the particle's path, even though it starts moving diagonally.
Visual Learning Try sketching the path and drawing velocity vectors at different points - you'll see how they're always tangent to the curve!
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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Explore detailed worked solutions for the A-Level Maths Paper 1 (2022). This resource covers key concepts such as differentiation, exponential functions, trigonometric equations, and area calculations, providing step-by-step explanations for each question. Ideal for students preparing for exams.
MathsExplore key concepts in kinematics, including average velocity, displacement, and acceleration. This summary covers essential calculations and examples from the first lesson of A-level Mechanics, focusing on displacement-time graphs and their implications in motion analysis. Ideal for Edexcel A-level Maths students.
MathsExplore key concepts of projectile motion including final velocity, time of flight, and trajectory analysis. This comprehensive summary covers both vertical and horizontal components, providing essential calculations and examples for A Level Maths (AQA). Perfect for mastering mechanics in physics.
MathsComprehensive overview of Year 1 Mechanics covering key concepts such as modelling assumptions, constant and variable acceleration, forces, motion, and the application of SUVAT equations. Ideal for AS level students seeking to reinforce their understanding of fundamental mechanics principles.
MathsComprehensive Year 1 Mechanics study material covering key concepts such as uniform motion, acceleration, Newton's Laws of Motion, kinematic equations, and free-body diagrams. Ideal for A-level Edexcel students, this resource provides essential insights into motion, forces, and mechanical systems, ensuring a solid foundation in Newtonian Mechanics.
MathsQuotes from every main character
English LiteratureApp Store
Google 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 S
iOS 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 Klich
Android 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.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
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 S
iOS 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 Klich
Android 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.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Kinematics is all about tracking how objects move through space and time - something you experience every day when you're walking, driving, or even just watching a ball fly through the air. This topic connects displacement, velocity, and acceleration using... Show more
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Think of kinematics like a GPS system for physics - it tracks position, velocity, and acceleration of moving objects. These three concepts are connected through differentiation and integration, making calculus your best mate for solving motion problems.
Displacement (r) tells you where a particle is located relative to a starting point. You can find this using coordinates (x, y) and calculate the distance with |r| = √. The displacement vector r is always underlined to show it's a vector quantity.
Velocity (v) shows how quickly position changes over time. You get this by differentiating displacement: v = dr/dt = (ẋ, ẏ). The speed is just the magnitude: |v| = √(ẋ² + ẏ²). Remember, velocity has direction whilst speed doesn't.
Acceleration (a) measures how velocity changes over time. It's the second derivative of displacement: a = dv/dt = d²r/dt² = (ẍ, ÿ). This tells you if something's speeding up, slowing down, or changing direction.
Quick Tip: Use dot notation (ẋ, ẏ) for first derivatives and double dots (ẍ, ÿ) for second derivatives - it's much faster than writing d/dt every time!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When two particles are moving through space, you can predict exactly when and where they'll cross paths by setting their position vectors equal to each other. It's like solving a collision problem in a video game!
For the example with particles E and F, you need to match up the i and j components separately. Set the i coefficients equal: -11 + 5t = 13 + t, which gives you t = 6s. Then do the same for j components: 23 - 3t = -7 + 2t, which also gives t = 6s.
Since both calculations give the same time, the particles definitely meet at t = 6 seconds. If you'd got different times, it would mean the particles never actually cross paths - they'd just pass by each other at different heights or distances.
To find the meeting position, substitute t = 6 back into either position vector. You should get the same answer from both particles: m. This confirms your calculation is spot on!
Pro Tip: Always check your answer by substituting the time into both position vectors - if they don't match, you've made an error somewhere!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Finding when particles are closest together or a specific distance apart involves working with relative displacement - basically tracking one particle from the perspective of the other. This technique is brilliant for collision avoidance problems!
Start by finding the relative displacement: rₐ - rᵦ. This gives you a single vector equation that describes how far apart the particles are. In the example, this becomes i + j, which represents the separation between particles A and B.
The distance squared is d² = ² + ² = 2t² - 20t + 52. For part (a), set d² = 100 and solve the quadratic to get t = 2s and t = 12s. The particles are 10m apart at both these times!
For part (b), find when particles are closest by differentiating d² and setting it to zero: d(d²)/dt = 4t - 20 = 0, giving t = 5s. At this moment, the particles are as close as they'll ever get during their motion.
Remember: When finding minimum distance, you're looking for when the rate of change of distance squared equals zero - that's your closest approach!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Understanding motion in 2D means tracking both x and y coordinates as they change over time. Think of it like plotting a character's movement across a game map - you need both horizontal and vertical positions to know exactly where they are.
The position vector r = (x(t), y(t)) gives you coordinates at any time t. In the example, r = traces out a curved path as t goes from 0 to 4. You can literally plot this on a graph to see the particle's journey!
Velocity vectors act like tangent lines to the path, showing the direction of movement at each instant. When t = 3, v = (6, -1), and when t = 4, v = (8, -1). Notice how these vectors point along the curve's direction at those specific moments.
The acceleration vector a = (2, 0) stays constant throughout this motion, always pointing horizontally to the right. This constant rightward acceleration gradually curves the particle's path, even though it starts moving diagonally.
Visual Learning: Try sketching the path and drawing velocity vectors at different points - you'll see how they're always tangent to the curve!
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
6
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Explore detailed worked solutions for the A-Level Maths Paper 1 (2022). This resource covers key concepts such as differentiation, exponential functions, trigonometric equations, and area calculations, providing step-by-step explanations for each question. Ideal for students preparing for exams.
MathsExplore key concepts in kinematics, including average velocity, displacement, and acceleration. This summary covers essential calculations and examples from the first lesson of A-level Mechanics, focusing on displacement-time graphs and their implications in motion analysis. Ideal for Edexcel A-level Maths students.
MathsExplore key concepts of projectile motion including final velocity, time of flight, and trajectory analysis. This comprehensive summary covers both vertical and horizontal components, providing essential calculations and examples for A Level Maths (AQA). Perfect for mastering mechanics in physics.
MathsComprehensive overview of Year 1 Mechanics covering key concepts such as modelling assumptions, constant and variable acceleration, forces, motion, and the application of SUVAT equations. Ideal for AS level students seeking to reinforce their understanding of fundamental mechanics principles.
MathsComprehensive Year 1 Mechanics study material covering key concepts such as uniform motion, acceleration, Newton's Laws of Motion, kinematic equations, and free-body diagrams. Ideal for A-level Edexcel students, this resource provides essential insights into motion, forces, and mechanical systems, ensuring a solid foundation in Newtonian Mechanics.
MathsQuotes from every main character
English LiteratureApp Store
Google 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 S
iOS 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 Klich
Android 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.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
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 S
iOS 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 Klich
Android 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.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Sociology