How to calculate the vertical acceleration of a projectile

The vertical acceleration of a projectile near Earth’s surface is a_y = −g ≈ −9.81 m/s² (downward) when air resistance is negligible. If air resistance matters, add a drag term: for linear drag a_y = −g − (k/m) v_y, and for quadratic (realistic at most speeds) a_y = −g − (ρ C_d A / 2m) v v_y. Below is a clear guide to choosing the right model, applying it step by step, and checking your results.

What “vertical acceleration” means in projectile motion

Vertical acceleration is the time rate of change of the projectile’s vertical velocity component. Choose a coordinate system with +y upward: gravity points downward, so the gravitational contribution is −g. In the no-drag model, that’s the entire story: a_y is constant and equal to −g at all times, including at the top of the trajectory. With drag, a_y depends on the current velocity because the drag force is velocity-dependent and always opposes motion.

Case 1: No air resistance (uniform gravity)

When air resistance is negligible (small, dense objects; modest speeds; short distances), the vertical acceleration is constant and equal to −g. Use this simple procedure to calculate it and apply it to related kinematic questions.

1. Define the vertical axis +y upward and take g ≈ 9.81 m/s² (9.80665 m/s² by convention). Near Earth’s surface, g can be treated as constant.
2. Write the vertical acceleration: a_y = −g.
3. If you need velocity or position: v_y(t) = v_{y0} − g t and y(t) = y_0 + v_{y0} t − (1/2) g t².
4. At the apex (v_y = 0), note that a_y is still −g. Acceleration does not vanish at the top; only vertical velocity does.
5. Use consistent units (SI preferred): meters, seconds, kilograms.

In this idealized but common scenario, the vertical acceleration never changes during flight; only the vertical velocity and position evolve in time according to the standard kinematic equations.

Quick numerical example (no drag)

Launch speed 20.0 m/s at 30° above horizontal: v_{y0} = 20.0 sin(30°) = 10.0 m/s. The vertical acceleration at every instant is a_y = −9.81 m/s². Time to apex is when v_y = 0: t_top = v_{y0}/g ≈ 10.0/9.81 ≈ 1.02 s; at that instant, a_y remains −9.81 m/s².

Case 2: Including air resistance

When air resistance cannot be ignored (light objects, high speed, large cross-sectional area, long flight), include drag. The vertical acceleration includes both gravity and drag, which depends on velocity. Choose a drag model compatible with the speed and Reynolds number.

1. Gather parameters: mass m, air density ρ (≈ 1.225 kg/m³ at sea level), drag coefficient C_d, cross-sectional area A, and current velocity components (v_x, v_y).
2. Quadratic drag (most common above a few m/s): drag magnitude is D = (1/2) ρ C_d A v² with v = sqrt(v_x² + v_y²); the drag force vector is opposite velocity. Vertical acceleration: a_y = −g − (ρ C_d A / 2m) v v_y.
3. Linear drag (useful at very low speeds or in some fluids): F_drag,y = −k v_y, so a_y = −g − (k/m) v_y, where k has units N·s/m.
4. At the apex (v_y = 0), drag contributes zero vertically, so a_y = −g even with drag (assuming no wind and no buoyancy correction).
5. Terminal velocity (straight-down fall): set a_y = 0. For quadratic drag, v_t = −sqrt(2 m g / (ρ C_d A)). For linear drag, v_t = −m g / k.

With drag, vertical acceleration evolves in time because it depends on the instantaneous velocity; you typically compute it step-by-step (numerically) or solve the differential equations for special cases.

Worked snapshot (quadratic drag)

Suppose m = 0.145 kg (baseball), A = 0.0042 m², C_d = 0.35, ρ = 1.225 kg/m³. At some instant, v_x = 20 m/s and v_y = 5 m/s upward. Then v = sqrt(20² + 5²) ≈ 20.6 m/s. The drag factor is (ρ C_d A)/(2m) ≈ (1.225×0.35×0.0042)/(2×0.145) ≈ 0.0062 s/m. The vertical acceleration is a_y = −9.81 − 0.0062×20.6×5 ≈ −9.81 − 0.64 ≈ −10.45 m/s².

Estimating vertical acceleration from data

If you have position or velocity data (from a motion sensor, smartphone IMU, or high-speed video), you can estimate a_y empirically.

1. Collect y(t) or v_y(t) at known time intervals (Δt).
2. Compute acceleration from velocity: a_y(t_i) ≈ [v_y(t_{i+1}) − v_y(t_{i−1})]/(2Δt) for better noise handling; or from position using second differences.
3. Apply smoothing (e.g., a low-order Savitzky–Golay filter) if the data are noisy.
4. Compare the mean a_y in mid-flight to −g; deviations indicate drag, wind, sensor bias, or calibration issues.

This approach is practical for lab experiments and lets you see when the idealized model breaks down due to aerodynamic effects.

Common pitfalls and quick checks

Several frequent mistakes can lead to incorrect results; use the checkpoints below to avoid them.

• Sign convention errors: define +y upward and keep it consistent; gravity is then −g.
• Confusing velocity with acceleration: at the top, v_y = 0 but a_y ≠ 0.
• Using the wrong drag model: quadratic drag dominates for most sports projectiles; linear drag suits very slow motion.
• Ignoring units: always keep SI units, and double-check that constants match those units.
• Assuming g is exactly 9.81 everywhere: slight variations with location and altitude exist but are usually small (<1%).

Running these checks early will save time and keep your calculations physically consistent.

Edge cases and refinements

In specialized settings, refine your model for better accuracy.

• Altitude effects: g(h) ≈ g_0 (R/(R + h))²; for typical projectile heights, the change is negligible.
• Buoyancy: subtract ρ_air V g / m from g if the object is low density or very large.
• Non-vertical winds: add wind to the air-relative velocity before computing drag.
• Noninertial frames: on rotating platforms or in long-range trajectories, include Coriolis and centripetal terms; vertical acceleration then includes small additional components.

These corrections matter most for very light objects, high altitudes, long ranges, or precision engineering applications.

Units and typical constants

Keep these values and unit tips at hand during calculations.

• g ≈ 9.81 m/s² (standard near sea level), 9.80665 m/s² by convention.
• Air density ρ ≈ 1.225 kg/m³ at 15°C, sea level; adjust for temperature and altitude if needed.
• Drag coefficient C_d varies: sphere ~0.47 (can be lower or higher with spin/roughness), streamlined shapes much lower.
• Areas A in m²; masses m in kg; velocities in m/s; accelerations in m/s².

Consistent units prevent silent errors and make constants directly usable without extra conversion factors.

Summary

To calculate the vertical acceleration of a projectile, set a_y = −g for no-drag motion; with drag, use a_y = −g − (ρ C_d A / 2m) v v_y (quadratic) or a_y = −g − (k/m) v_y (linear). The acceleration remains downward at all times—even at the apex—and depends on instantaneous velocity only when aerodynamic forces are included. Choose the model that fits your speed and size regime, keep a consistent sign convention, and verify results with data when possible.