Loading 3D…
Now: vx / vy —Sideways drift (wind) —Playback —
Two motions at once
Twin ball: a second ball launched straight up (or dropped) with only the up-and-down part of the launch. 3D: drag sideways to turn the field.
The 2D simulator is the page’s original simulator: Free Launch, Compare Angles (15° to 75° at once), From Height and Hit Target, with graphs of y(x), y(t), vx,vy(t), |v|(t) and R vs θ, and planet buttons. The launch log collects every launch from the 3D lab in a table and a range-vs-angle graph.
Free launch — adjust angle & speed, then press Fire!
Every launch you make in the 3D Projectile Lab above is written here (the newest first). The graph shows range against launch angle for the launches that match your latest settings (speed, height, gravity, air and wind).
| # | Speed (m/s) | Angle (°) | Height (m) | World / air | Time of flight (s) | Range (m) | Max height (m) |
|---|---|---|---|---|---|---|---|
| No launches yet: press 🚀 Launch in the lab above. | |||||||
Projectile motion is the curved path of an object that is launched and then moves under gravity alone. Its sideways speed stays the same while gravity changes its up-and-down speed by 9.81 m/s every second on Earth, so the path is a parabola. On level ground with no air, a 45° launch goes farthest: 40.77 m at 20 m/s.
| If you… | the range… | because… |
|---|---|---|
| launch faster | grows a lot (2× speed → 4× range) | the ball moves faster sideways AND stays up longer |
| change the angle | is largest at 45°; 30° and 60° tie | a low throw lands too soon, a high throw moves forward too slowly |
| go to weaker gravity (Moon) | grows (about 6× on the Moon) | the ball takes longer to fall back down |
| launch from higher up | grows; the best angle drops below 45° | the ball gets extra time to fall |
| add air resistance | shrinks; the best angle drops below 45° | drag keeps slowing the ball |
Pick one answer for each question, then press Check my answers to see the explanations. New questions picks another 10 from a bank of 24.
The equations below are for level ground and no air unless stated. They are the “formula” values the 3D lab shows next to its measured values.
vx = v₀ cos θ vy0 = v₀ sin θ
At 20 m/s and 45°: vx = vy0 = 20 × 0.7071 = 14.14 m/s. With no air, vx never changes; vy changes by g every second.
x(t) = vx t y(t) = h₀ + vy0 t − ½ g t² vy(t) = vy0 − g t
T = 2 v₀ sin θ / g H = (v₀ sin θ)² / (2g) R = v₀² sin 2θ / g (h₀ = 0)
R is largest when sin 2θ = 1, so at θ = 45°. Because sin(180° − 2θ) = sin 2θ, the angles θ and 90° − θ give the same range. Eliminating t gives the path y = h₀ + x tan θ − g x² / (2 v₀² cos² θ): a downward-opening parabola.
vx = 24 cos 35° = 19.66 m/s; vy0 = 24 sin 35° = 13.77 m/s; T = 2(13.77)/9.81 = 2.81 s; H = 13.77²/(2 × 9.81) = 9.66 m; R = 24² sin 70° / 9.81 = 55.17 m. Set 24 m/s and 35° in the lab: it shows the same values, so the percent difference |measured − predicted| / predicted × 100 is 0% (small rounding differences under 0.5% are fine).
T = (vy0 + √(vy0² + 2 g h₀)) / g best angle: tan θ = v₀ / √(v₀² + 2 g h₀)
The best angle is below 45° because the ball already has extra time to fall. At 20 m/s from 20 m: θ = 35.4°, which ⭐ Best angle rounds to the slider step: 35.5°. Max height in the lab is measured from the ground, so it includes h₀: H = h₀ + vy0² / (2g).
The 3D lab uses quadratic drag, F = ½ ρ Cd A v², pointing against the ball’s motion through the air (the crosswind blows along the field’s width). Ball data: baseball Cd 0.35, radius 3.7 cm, 145 g; tennis ball 0.55, 3.35 cm, 57 g; ping-pong ball 0.5, 2.0 cm, 2.7 g. Air density: Earth 1.2 kg/m³, Mars 0.02, Jupiter 0.16 (1-bar level), Moon 0. The motion is stepped every 0.1–1 ms with the fourth-order Runge–Kutta method and the landing point is interpolated.
With drag the two motions are no longer independent: the drag depends on the total speed, so the sideways motion also slows the fall a little. That is why, with air on, the twin ball and the thrown ball no longer land together. At 20 m/s the best angles are 43.5° (baseball, 34.25 m), 41.5° (tennis ball, 25.81 m) and 35° (ping-pong ball, 9.98 m): drag keeps slowing the ball, so time spent high in the air costs range, and a lower, flatter launch goes farther. The 2D simulator in the tabs uses a simpler linear drag, F = −k v, on a 1 kg ball: v̇x = −k vx/m, v̇y = −g − k vy/m.
| Symbol | Quantity | SI unit |
|---|---|---|
| v₀ | launch speed | m/s |
| θ | launch angle above the horizontal | degree |
| h₀ | launch height above the ground | m |
| g | acceleration due to gravity | m/s² |
| R | range (horizontal distance) | m |
| H | maximum height | m |
| T | time of flight | s |
| k | linear drag coefficient (2D simulator) | kg/s |
NGSS MS-PS2-2: the change in motion depends on the force (gravity changes only the vertical motion; no sideways force, no sideways change). HS-PS2-1 is limited to one-dimensional motion in its assessment boundary: use the vertical motion (the vy–t graph in the 2D simulator: a straight line with slope −9.81 m/s²) as the one-dimensional evidence; the full 2D projectile is an extension. AP Physics 1 covers projectile motion in its kinematics unit (motion in two dimensions).
References: Halliday, Resnick & Walker, Fundamentals of Physics, 10th ed., ch. 4; Serway & Jewett, 8th ed., §4.3; Kleppner & Kolenkow, 2nd ed., §3.3; misconception research: Halloun & Hestenes (1985), Am. J. Phys. 53, 1043; Clement (1982), Am. J. Phys. 50, 66.
One lesson (about 45 minutes) + homework: start with the Drop race as a prediction (hands up: which lands first?), then let students do the 7 missions beside the lab. In class: worksheet parts A–B (angle vs range at 16 m/s, faster throws) and the 10-question quiz; parts C–F (the race, other worlds, a target, explain) as homework.
Exact values by tap: every slider has − / + buttons of one step (0.5 m/s, 0.5°, 0.5 m), so students on Chromebooks, iPads and phones can set the values a worksheet asks for. The readouts change as soon as a value changes; 🚀 Launch replays the flight.
Model notes: the 3D lab is exact for a vacuum (the “formula” value next to each readout agrees). With air it uses quadratic drag with real ball sizes and masses, stepped every 0.1–1 ms. Max height includes the launch height. A target counts as hit within 0.5 m. Jupiter’s 24.79 m/s² is a textbook value.
Two labs, one page: the 3D lab at the top, and the original 2D simulator (compare angles, from height, hit target, graphs of y(x), y(t), vx,vy(t), |v|(t) and R vs θ) in the tab box right under it. The launch log tab records every launch for a range-vs-angle graph.
High school: the “Go deeper” section has the components, the kinematics equations, a worked example with percent difference, launch height and air resistance. A printable two-level lab pack with answer keys is linked in the For teachers box at the bottom of the page.
Standards: NGSS MS-PS2-2 (gravity changes only the vertical motion; one variable at a time); HS-PS2-1 for the vertical motion (vy–t graph) and AP Physics 1 kinematics in Go deeper. Full 2D projectile motion goes beyond the NGSS high school assessment boundary (1D), so there it is an extension.
| Time | Step | What students do | Go to |
|---|---|---|---|
| 5 min | Hook | Drop race: predict which ball lands first, then press it | Simulator |
| 15 min | Explore | Missions 1–4: race, best angle, twin angles, twice as fast | Missions |
| 10 min | Investigate | Worksheet A–B: angle vs range at 16 m/s; plot range against angle (C–F as homework) | Launch log |
| 5 min | Apply | Target challenge and the Moon mission | Target |
| 10 min | Check | Quiz: 10 questions from a bank of 24 | Quiz |
Worksheet: a print-out for students and a separate answer key page for the teacher (opens the print dialog; choose "Save as PDF" to keep a file).
Projectile motion is the motion of an object that is launched and then moves under gravity alone, like a thrown ball or water from a hose. It is two motions at once: a steady sideways motion and an up-and-down motion that gravity changes by 9.81 m/s every second (on Earth). Together they make a curved path called a parabola.
Key takeaway: Steady sideways + changing up-and-down = a parabola.On level ground with no air, 45° goes farthest. A low throw moves forward fast but lands too soon; a high throw stays up long but moves forward slowly. 45° is the best balance: at 20 m/s it lands 40.77 m away. From a height, or with air resistance, the best angle is lower.
Key takeaway: 45° is best only on level ground with no air.Yes, if the air can be ignored. Both start with zero up-and-down speed, and gravity pulls both down in the same way, so they fall side by side and land together. The thrown ball just lands farther away. Press 🏁 Drop race in the lab: from 20 m up, both land after 2.02 s at any throwing speed.
Key takeaway: Sideways motion does not change how fast an object falls.At the top only the vertical (up-and-down) velocity is zero. The horizontal velocity is still there, so the ball keeps moving forward: for a 20 m/s, 45° throw its speed at the top is 14.14 m/s, not zero. Only a ball thrown straight up (90°) really stops for a moment.
Key takeaway: At the top vy = 0 but vx is unchanged.Yes, on level ground with no air. Angles that add up to 90° land at the same spot: at 20 m/s, 30° and 60° both land 35.31 m away. The 60° throw goes higher (15.29 m instead of 5.10 m) and stays up longer, but it moves forward more slowly. (High school: why this works is in Go deeper.)
Key takeaway: Complementary angles tie in range; the steeper one flies higher and longer.Not without air: in a vacuum, mass cancels out and every object launched the same way follows the same path (on the Moon, a baseball and a ping-pong ball both land 246.91 m away at 20 m/s and 45°). With air, a ball that is heavy for its size is slowed less: on Earth at 20 m/s a baseball’s best throw lands 34.25 m away, a ping-pong ball’s only 9.98 m.
Key takeaway: Mass matters only through air resistance.Air drag slows the ball the whole time, so the range and the maximum height get smaller, and the path is no longer a symmetric parabola: the way down is steeper than the way up. The best angle drops below 45° (43.5° for a baseball, 41.5° for a tennis ball and 35° for a ping-pong ball at 20 m/s in the 3D lab). Choose an air setting in the lab to compare with the dashed no-air path.
Key takeaway: Air makes the path shorter, lower and lopsided, and lowers the best angle.On level ground with no air: R = v₀² sin 2θ / g. At 20 m/s and 45° on Earth, R = 400 × 1 / 9.81 = 40.77 m. The time of flight is T = 2 v₀ sin θ / g and the maximum height H = (v₀ sin θ)² / (2g). From a height h₀ use T = (vy0 + √(vy0² + 2 g h₀)) / g and R = v₀ cos θ × T. Go deeper has a worked example.
Key takeaway: R = v₀² sin 2θ / g on level ground.Everywhere something is thrown, kicked, hit or jumps: a basketball free throw, a soccer kick, a javelin, a golf ball, a long jump, water from a hose or a fountain, a frog’s jump or a thrown stone. Fast or light objects (a golf ball, a badminton shuttle) are strongly affected by air, so their real paths are shorter than the no-air prediction.
Key takeaway: Any launched object that then moves under gravity is a projectile.The 3D lab launches a ball from a cannon with the speed, angle and height you choose, on Earth, the Moon, Mars or Jupiter, with or without air. It shows the path, equal-time dots with their ground shadows, the sideways (red) and up-and-down (green) speed arrows, and the time of flight, range and maximum height next to the no-air formula. The 2D simulator in the tabs adds Compare Angles, From Height and Hit Target modes and graphs of position and velocity against time.
Key takeaway: Two linked labs: the 3D launcher and the 2D simulator with graphs.