DallaTools

Projectile Motion Calculator

Enter initial speed (m/s), launch angle (°), and initial height (m). Use default g = 9.81 m/s² or set your own. Results update instantly.

Advanced options
Values at time t
x(t)
14.142
y(t)
9.237
vᵧ(t)
4.332
|v(t)|
14.791
Still airborne
Horizontal component vₓ
14.142
Vertical component vᵧ
14.142
Time of flight
2.883
Range
40.775
Max height
10.194
Time to apex
1.442
Horizontal position at apex
20.387

Projectile Motion: Complete Guide (No Air Resistance)

This calculator uses the standard constant-gravity model (g > 0) and neglects air drag. Motion splits into independent components: horizontal at constant vₓ, and vertical with constant acceleration −g. Position at time t: x(t) = vₓ · t and y(t) = h₀ + vᵧ · t − ½ g t², with vₓ = v₀ cos θ and vᵧ = v₀ sin θ.

How the Calculator Works (Step by Step)

  1. Unit normalization: speeds → m/s, lengths → m, gravity → m/s², angles → radians.
  2. Components: vₓ = v₀ cos θ, vᵧ = v₀ sin θ (or use advanced vₓ, vᵧ if provided).
  3. Time of flight: solve y(T)=0 to get T = (vᵧ + √(vᵧ² + 2 g h₀)) / g.
  4. Range: R = vₓ · T (a horizontal displacement; use |vₓ| · T if you prefer distance).
  5. Apex: tₐ = clamp(vᵧ/g, 0, T); then xₐ = vₓ · tₐ and h_max = y(tₐ).
  6. At a chosen time: x(t), y(t), vᵧ(t)=vᵧ−g t, and speed |v| = √(vₓ² + vᵧ(t)²).

Key Formulas

  • Time of flight: T = (vᵧ + √(vᵧ² + 2 g h₀)) / g.
  • Range (displacement): R = vₓ · T. For unsigned “distance,” use |vₓ| · T.
  • Time to apex: tₐ = clamp(vᵧ/g, 0, T).
  • Horizontal position at apex: xₐ = vₓ · tₐ.
  • Maximum height: h_max = h₀ + vᵧ · tₐ − ½ g tₐ² (this equals h₀ if vᵧ ≤ 0).
  • Level ground (h₀ = 0): T = 2 vᵧ / g, R = (v₀² sin 2θ)/g.

Worked Example (SI Units)

v₀ = 20 m/s, θ = 45°, h₀ = 0, g = 9.81 m/s² ⇒ vₓ = vᵧ ≈ 14.142 m/s. Results: T ≈ 2.88 s, R ≈ 40.8 m, tₐ ≈ 1.44 s, h_max ≈ 10.2 m.

When is 45° Optimal?

With h₀ = 0 and no air drag, maximum range occurs at θ = 45°. For h₀ > 0 the optimal angle is less than 45° because extra drop time from elevation increases horizontal travel even at shallower angles.

Units & Conversions

  • Angles: convert degrees → radians for trig (the tool does this automatically when “deg” is selected).
  • Speed: km/h → m/s divide by 3.6; mph → m/s divide by 2.23693629; ft/s → m/s divide by 3.28084.
  • Length: ft → m divide by 3.28084. Gravity: ft/s² → m/s² divide by 3.28084.
  • The calculator converts to SI internally and then formats outputs in your selected units.

Special Cases & Edge Handling

  • Horizontal launch (θ = 0° ⇒ vᵧ = 0): tₐ = 0 and h_max = h₀; descent follows free-fall.
  • Downward launch (vᵧ ≤ 0): apex before t=0 is ignored; we clamp tₐ to [0, T] so displayed apex is physically meaningful.
  • Large initial height (h₀ > 0): flight time and range increase even for sub-45° angles.

Common Pitfalls

  • Forgetting degree→radian conversion when doing manual trig.
  • Mixing units (km/h vs m/s, ft vs m) or gravity units.
  • Assuming validity at very high speeds where aerodynamic drag dominates and this model breaks down.

FAQ

Is “range” signed? By definition here it’s horizontal displacement, R = vₓ T, which can be negative if vₓ < 0. Use |R| for distance.

Why clamp tₐ? The turning point at vᵧ/g might lie before launch (t < 0) or after landing (t > T). Clamping keeps apex within the actual flight.

Can I input components directly? Yes—enable the advanced toggle and enter vₓ, vᵧ in your chosen speed units.

Practice Problems

  1. Ball at 18 m/s, θ = 30°, h₀ = 0: compute T, R, h_max.
  2. Projectile from a 12 m cliff, 22 m/s at 40°: find time of flight and range.

Note: this model ignores aerodynamic drag, spin, and wind. For high-speed or long-range applications, use a drag-aware trajectory model.