When tackling kinematics problems, a quick way to decide which equation to use is to look carefully at which variables are given and which one you need to find. The four equations involve only five quantities — $u$ (initial velocity), $v$ (final velocity), $a$ (acceleration), $t$ (time), and $s$(displacement). Each equation links four of them, leaving one out. So if, for example, time $t$ is not mentioned anywhere in the problem, the best choice is usually the equation $v^2 = u^2 + 2as$, since that one does not involve $t$. By matching the known and unknown quantities against the “missing variable” in each equation, you can quickly narrow down the correct equation to apply.
This simulation offers a clear and interactive way to explore the motion of a ball bouncing on the ground, highlighting how displacement, velocity, and acceleration change over time. On the left, you’ll see the animation of the ball with vectors showing its position (green), velocity (pink), and acceleration (blue). The sliders at the top allow you to adjust the starting height, the percentage of energy lost on each bounce, and whether air resistance is included. You can pause, reset, or let the motion run continuously, while the time slider doubles as a scrubber when the simulation is paused.
On the right, the three graphs display how each physical quantity varies with time. The position–time graph shows the ball’s vertical displacement, always measured relative to the lowest point of its center of mass. The velocity–time graph alternates between negative and positive values, reflecting the downward and upward motion during each bounce, while the acceleration–time graph remains mostly constant at –g, with spikes at the moment of collision. Together, the animation and graphs help link the visual motion with the quantitative data, reinforcing the relationships between these variables.
The underlying theory follows Newton’s laws of motion. The ball accelerates downwards under gravity until it collides with the ground, where it loses some energy depending on the restitution factor. This is why the bounce height diminishes over time. The velocity vector shows not only the speed but also the direction of motion, while the acceleration vector indicates that gravity always acts downward, regardless of whether the ball is rising or falling. By adjusting energy loss during each collision and air resistance, you can model more realistic scenarios and see how dissipative forces affect motion, making this a powerful tool to visualize the physics of bouncing objects.
This simulation lets you watch two equal-size spheres (light and heavy) fall, while you see their free-body diagrams (FBDs) and a velocity–time graph update in real time.
Press Start to run and Pause to discuss a moment in time. Reset restarts from rest. Use Zoom to read the force vectors clearly. Toggle Air Resistance to compare an idealised fall (no drag) with a more realistic one (drag on). The small info panel shows the current speeds and, when drag is on, each sphere’s terminal velocity.
What to look for
Without air resistance: Each FBD shows only weight downward. Acceleration is constant at ggg, so the velocity–time graph is a straight line from the origin for both spheres (same slope, because mass doesn’t matter when no drag acts).
With air resistance: A drag arrow appears upward and grows with speed. The heavy sphere’s velocity rises faster at first (its weight is larger), but both curves flatten as drag increases, and the acceleration vector shrinks toward zero. Dotted segments indicate when the two curves overlap closely.
Theory in one breath
In air, we model drag as proportional to speed: $F_\text{drag}=kv$
Net force is $ma=mg−kv \quad\Rightarrow a=g−\dfrac{k}{m}v$
As v grows, the term $\dfrac{k}{m}v$ eats into $g$, so acceleration falls.
Terminal velocity happens when forces balance: $mg=kv$, so $v_t=\dfrac{mg}{k}$
Heavier mass ⇒ larger $v_t$. That’s why the heavy sphere ultimately settles at a higher speed and takes longer to level off. With drag off, the model is simply $a=g$ and $v=gt$.
How to teach with it (fast)
Start with Air Resistance off: Pause after a second—ask why both lines match and why only weight appears on the FBDs.
Turn Air Resistance on: Run, then pause midway. What changed in the FBDs? Why is acceleration smaller now?
Let it run until the acceleration vectors nearly vanish: connect “flat graph” with “balanced forces,” then read off different terminal velocities.
That’s it: start, pause, notice which vector changed, and link the picture to the equation.
Understanding motion in physics often involves analyzing displacement, velocity, and acceleration graphs. With the interactive GeoGebra graph at this link, you can dynamically explore how these concepts are connected.
How It Works
This interactive simulation lets you visualize an object’s motion and its corresponding displacement-time, velocity-time, and acceleration-time graphs. You can interact with the model in two key ways:
Adjust Initial Conditions:
Move the dots on the graph to change the starting displacement, velocity, or acceleration.
Observe how these changes influence the overall motion of the object.
Use the Slider to Animate Motion:
Slide through time to see how the object moves along its path.
Watch the displacement vector, velocity vector, and acceleration vector update in real time.
Key Observations
When displacement changes, the velocity and acceleration graphs adjust accordingly.
A constant acceleration results in a straight-line velocity graph and a quadratic displacement graph.
Negative acceleration (deceleration) slows the object down and can cause direction reversals.
If velocity is constant, the displacement graph is linear, and acceleration remains at zero.
Why This is Useful
This GeoGebra tool is perfect for students and educators looking to build intuition about kinematics. Instead of just solving equations, you get a visual and hands-on way to see the relationships between these key motion variables.
Try it out yourself and experiment with different conditions to deepen your understanding of motion!
This week, I conducted a lesson on motion for my IP3 class using a simple yet effective tool: a simulated ticker tape timer. The objective was to help students develop an intuitive understanding of uniform and non-uniform motion by actively engaging in an experiment.
Introduction to the Ticker Tape Timer
To kickstart the lesson, I showed my students a YouTube video that explains how a ticker tape timer works:
This video provided a visual demonstration of how a ticker tape timer marks regular intervals on a moving strip of paper, allowing us to analyze motion quantitatively.
Hands-On Experiment: Simulating a Ticker Tape Timer
After the video, I had students pair up for a hands-on activity. Instead of using an actual ticker tape timer, we simulated the process using paper strips cut from A3-sized sheets. Each pair had one student act as the “moving arm,” responsible for placing dots on the strip, while the other played the role of the “puller,” responsible for pulling the paper strip at different speeds.
To ensure a consistent time interval between each dot, I used a Metronome App that I created:
This app produces a steady rhythm at 120 beats per minute, meaning that the interval between each beep (and consequently each dot) is 0.5 seconds. To improve accuracy, the student acting as the moving arm was instructed to close their eyes and focus solely on the beep.
Step 1: Recording Uniform Motion
In the first trial, the puller was asked to pull the paper at a constant rate. As the paper moved steadily, the moving arm marked dots at regular intervals based on the metronome beat. After completing the trial, students used a ruler to measure the distances between successive dots. Since the time interval was fixed, they could easily calculate the speed of the paper by using:
Step 2: Recording Accelerated Motion
Next, the students switched roles. This time, the new puller was asked to gradually increase the speed of the paper. As expected, the spacing between dots increased progressively, providing a clear visual representation of acceleration. This led to discussions on how motion can be analyzed using dot patterns and how acceleration differs from uniform motion.
Reflections and Key Takeaways
This activity was highly effective in reinforcing key motion concepts. Since we do not have an actual ticker tape machine, it allowed students to engage in a hands-on simulation while visually and physically experience motion rather than just reading about it.
Next Steps
To extend this lesson, I plan to introduce velocity-time graphs and have students plot their measured speeds to analyze changes in motion further. Additionally, incorporating digital tools like video analysis with Tracker software could help reinforce these concepts further.
If you have any feedback or suggestions, feel free to share them in the comments below!
I was experimenting with using generative AI to create an interactive graph that could be used to amend the animation of a moving particle, for the topic of kinematics. Students are able to move the four points on the velocity time graph to manipulate the movement. I kept the graph to straight lines between each point to keep things simple.
The vertical axis toggles between displacement and velocity. This will be yet another way for students to learn about how the velocity-time graph affects motion. I have found that many students are confused between displacement and velocity. The app’s ability for them to vary the velocity graph and then make predictions of the resulting displacement graph and the movement should be worth the effort.