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Kinematics: Motion in 1D & 2D — Visual Lab
Module 1. Pick a launch speed, an angle, and the strength of gravity — then press Launch and watch a ball trace its real parabolic arc. The horizontal motion runs steady while gravity quietly bends the vertical, and the range, peak height, and flight time you read off are exactly what the equations predict.
Two-dimensional motion is just two one-dimensional problems running at once. Split the launch velocity into a horizontal part \(v_x = v\cos\theta\) and a vertical part \(v_y = v\sin\theta\). Sideways, nothing pushes the ball, so \(v_x\) never changes. Upward, gravity steadily removes speed until the ball stops rising, then hands it back on the way down. Put the two together and you get a parabola. Drive the sandbox below and keep the arc, the vectors, and the algebra in perfect agreement.
Projectile Sandbox
Set the launch speed \(v\), the angle \(\theta\), and gravity \(g\), then press Launch. The ball flies its true parabola while the live readout reports range, max height, and time of flight; the velocity vector at the ball splits into its constant horizontal part and its changing vertical part.
What you're seeing
- The arc is a parabola. The faint dashed curve is the whole future path; the bold curve is how far the ball has actually flown so far.
- The velocity vector points along the motion. Split on, it shows the horizontal component \(v_x = v\cos\theta\) (always the same length) and the vertical component \(v_y = v\sin\theta - g t\) (shrinking, zero at the top, then growing downward).
- Apex and landing are marked. The top point is the max height \(H\); the dot on the ground is the range \(R\).
- Gravity is a dial. Slide \(g\) from the Moon to Jupiter and watch the same launch land much farther or much shorter.
- The numbers come from the formulas. The readout and the formula panel always agree with the picture — the animation is the equations.
Try this
- Hunt for the farthest shot. Hold the speed fixed and sweep the angle. The range peaks at 45° — convince yourself with the readout.
- Compare complementary angles. Launch at 30°, note the range, then launch at 60°. Same range, different shaped arc — that's \(\sin(2\theta)\) at work.
- Watch \(v_y\) die at the top. Pause near the apex: the vertical arrow vanishes while the horizontal arrow is unchanged. All the speed there is sideways.
- Change gravity, keep the launch. Drop \(g\) to the Moon's \(1.6\) and replay — the ball hangs far longer and flies much farther on the very same speed and angle.
Worked Examples
Two moves this module asks for most: resolving a launch into components, and chaining those components into range, height, and time.
Example 1 — Resolve a launch into components
A ball is launched at \(20\) m/s, \(30^\circ\) above the horizontal. Find \(v_x\) and \(v_y\).
- Horizontal: use cosine. \(v_x = v\cos\theta = 20\cos 30^\circ\).
- Evaluate. \(\cos 30^\circ = \tfrac{\sqrt3}{2}\approx 0.866\), so \(v_x \approx 20(0.866) = 17.32\) m/s.
- Vertical: use sine. \(v_y = v\sin\theta = 20\sin 30^\circ\).
- Evaluate. \(\sin 30^\circ = \tfrac12\), so \(v_y = 20(0.5) = 10\) m/s.
Example 2 — Range, height, and time from a launch
A projectile leaves level ground at \(30\) m/s, \(40^\circ\), with \(g = 9.8\) m/s². Find its time of flight, max height, and range.
- Vertical launch speed. \(v_y = 30\sin 40^\circ \approx 30(0.643) = 19.28\) m/s.
- Time of flight. \(T = \dfrac{2v_y}{g} = \dfrac{2(19.28)}{9.8} \approx 3.93\) s (up and back down).
- Max height. \(H = \dfrac{v_y^2}{2g} = \dfrac{19.28^2}{2(9.8)} \approx 18.97\) m.
- Range. \(R = \dfrac{v^2\sin(2\theta)}{g} = \dfrac{30^2\sin 80^\circ}{9.8} \approx 90.45\) m.
Key Vocabulary
The precise words physicists use to describe what the lab is doing.
The straight-line change in position, with direction. Different from distance, which doesn't care about direction.
How fast position changes and in what direction: a vector. Speed is just its size, with the direction stripped off.
How fast velocity changes. In projectile motion it points straight down with size \(g\) — gravity, and nothing else.
A vector split along the axes: \(v_x = v\cos\theta\) and \(v_y = v\sin\theta\). The two move independently of each other.
An object moving under gravity alone after launch. Its path is a parabola when air resistance is ignored.
Range \(R\) is the horizontal distance covered before landing; time of flight \(T\) is how long it stays airborne.
Where this fits
This is enrichment physics — not a TEKS-graded course and not tied to a state End-of-Course exam. The kinematics here is the standard algebra-based treatment students meet in a first physics course; it leans directly on the trigonometry and equation-solving from the math pathway. Use it to stretch and explore, not for a grade.
Constant-velocity & constant-acceleration models · vectors and their components · two-dimensional projectile motion over level ground, idealized with no air resistance.
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Physics enrichment content — not part of the TEA Bluebonnet mathematics sequence and not a TEKS-graded course. Offered as optional stretch material for curious scholars. Non-commercial classroom use.