Projectile Motion Virtual Lab

A projectile is an object that, once launched, moves under the influence of gravity alone. A thrown ball, a kicked football, a jet of water and a stone skimmed off a cliff all follow curved paths that look similar, and one compact set of ideas explains them all. The central idea is that the horizontal motion and the vertical motion can be analysed separately, then recombined.

This page works in three layers. Material marked Foundation needs only basic algebra and the idea of speed. Core material matches typical A-level, IB Diploma, AP Physics 1, WASSCE and Cambridge AS/A Level courses and uses vectors, trigonometry and the equations of constant acceleration. Extension material (marked clearly) goes towards IB HL, AP Physics C: Mechanics and first-year university, and uses calculus ideas and optimisation.

Use the interactive simulation alongside these notes. Its default setting is launch speed u = 20 m/s, launch angle θ = 45° and launch height h = 0, on Earth with g = 9.81 m/s². Before you change anything, predict what will happen, then test your prediction.

Learning outcomes

Foundation

• Describe a projectile and state that only gravity acts on it when air resistance is ignored.
• Explain that a projectile keeps moving sideways at a steady speed while it speeds up and slows down vertically.
• Sketch the path of a projectile and recognise it as a curve (a parabola in the ideal case).
• Use speed = distance ÷ time for the horizontal motion and v = u + at for the vertical motion in simple one-step problems.
• Explain why a lower value of g (for example on the Moon) gives a longer flight.

Core (A-level, IB, AP, WAEC)

• Resolve a launch velocity into horizontal and vertical components, and recombine components into a speed and a direction.
• Apply the equations of constant acceleration separately to each direction, with a consistent sign convention.
• Calculate time of flight, range, maximum height and impact velocity for launches from level ground and from a height.
• Solve problems that lead to a quadratic equation in time.
• Derive and use R = u² sin 2θ / g on level ground, and explain complementary angles.
• Interpret displacement-time, velocity-time and path graphs for a projectile.
• Describe qualitatively how air resistance changes range, height and path shape.
• Plan an investigation, estimate uncertainties, and use a graph of R against sin 2θ to find g.

Extension (HL, AP C, university)

• Derive the trajectory equation y(x) and the optimum launch angle from a height, tan θ = u / √(u² + 2gh).
• Use energy conservation to find impact speed without finding the time.
• Write down the equation of motion with quadratic drag, define terminal velocity, and describe numerical integration (Euler and fourth-order Runge-Kutta).
• Describe the safety parabola (envelope), projectiles on an incline, and the Magnus effect.

What counts as a projectile

A projectile is any object moving through the air (or a vacuum) after being launched, where the only force worth considering is gravity. The launch itself (the throw, kick or firing) is not part of the projectile motion: the analysis begins at the instant the object leaves the launcher, with a known initial velocity.

Vectors and components

Velocity is a vector: it has a size (speed) and a direction. A vector can be replaced by two perpendicular components. If the launch speed is u at angle θ above the horizontal, then the horizontal component is u_x = u cos θ and the vertical component is u_y = u sin θ.

Independence of horizontal and vertical motion

Independence does not mean the two motions are unrelated. They share the same clock: the time of flight is set by the vertical motion, and the horizontal distance covered in that time is then horizontal speed × time. Two balls, one dropped and one launched sideways from the same height at the same moment, hit the ground together, because their vertical motions are identical.

Constant acceleration

Near the surface of a planet, gravity gives every object the same downward acceleration g (9.81 m/s² on Earth in these notes), regardless of mass, once air resistance is neglected. So:

• Horizontal: acceleration a_x = 0, so v_x is constant.
• Vertical: acceleration a_y = −g (with upward taken as positive), so v_y changes by 9.81 m/s every second.

Because the acceleration is constant in each direction, the equations of constant acceleration (SUVAT) apply to each direction separately. They do not apply to the combined motion along the curved path.

Ideal assumptions

The standard model assumes: (1) the projectile is a point mass, so spin and size do not matter; (2) air resistance is zero; (3) g is uniform in size and direction over the whole flight (the ground is flat and the height gained is tiny compared with the planet's radius); (4) no wind; (5) the launch and landing are at known positions, usually with ground at y = 0. The simulation uses these assumptions unless you turn on drag.

C1. The equations of constant acceleration

For motion in a straight line with constant acceleration a, initial velocity u, final velocity v, displacement s and time t:

v = u + at

s = ut + ½at²

v² = u² + 2as

s = ½(u + v)t

Here s is displacement (a vector quantity with sign), not total distance. These are derived in every mechanics course by integrating constant acceleration; in this lab we simply use them, one direction at a time.

C2. Component resolution and the two motions

Take the launch point at height h above ground, origin of x directly below it, upward positive. At time t after launch:

v_x = u cos θ (constant)    x = (u cos θ) t

v_y = u sin θ − gt    y = h + (u sin θ) t − ½gt²

The speed at any time is v = √(v_x² + v_y²), and the direction below or above the horizontal is given by tan φ = |v_y| / v_x.

C3. Trajectory equation y(x)

1. From the horizontal equation, t = x / (u cos θ).
2. Substitute into the vertical position: y = h + u sin θ × x / (u cos θ) − ½g [x / (u cos θ)]².
3. Simplify, using sin θ / cos θ = tan θ:

y = h + x tan θ − gx² / (2u² cos²θ)

Since 1 / cos²θ = 1 + tan²θ, an equivalent form is y = h + x tan θ − gx²(1 + tan²θ) / (2u²). This is a quadratic in x with a negative coefficient on x², so the path is a downward-opening parabola.

C4. Time of flight from a height

The projectile lands when y = 0:

1. 0 = h + u sin θ t − ½gt², which rearranges to ½gt² − u sin θ t − h = 0.
2. Quadratic formula: t = [u sin θ ± √(u² sin²θ + 2gh)] / g.
3. The minus sign gives a negative time (before launch), which has no physical meaning here. So:

T = [u sin θ + √(u² sin²θ + 2gh)] / g

For h = 0 this gives T = 2u sin θ / g. For a horizontal launch (θ = 0) it gives T = √(2h/g).

C5. Maximum height

At the top v_y = 0. Using v_y² = u_y² − 2g(y − h) with v_y = 0:

y_max = h + u² sin²θ / (2g)

The time to reach the top is t_top = u sin θ / g.

C6. Range from a height

Range is the horizontal distance at landing, R = u cos θ × T:

R = (u cos θ / g) [u sin θ + √(u² sin²θ + 2gh)]

C7. Range on level ground and complementary angles

Set h = 0, so T = 2u sin θ / g and

R = u cos θ × 2u sin θ / g = u² sin 2θ / g

using the double-angle identity 2 sin θ cos θ = sin 2θ. The range is largest when sin 2θ = 1, so θ = 45°, giving R_max = u²/g.

Symmetry. Because sin(180° − 2θ) = sin 2θ, the angles θ and (90° − θ) give the same range, since 2(90° − θ) = 180° − 2θ. These are complementary angles (they add to 90°). The steeper path of the pair is higher and takes longer, but covers the same ground distance.

C8. Optimum angle from a height (extension)

For a launch from height h at fixed speed u, the best angle is less than 45°. Here is a derivation that avoids calculus. Let k = tan θ. The trajectory equation, with landing at (R, 0), becomes

0 = h + Rk − gR²(1 + k²) / (2u²)

1. Rearrange as a quadratic in k: (gR² / 2u²) k² − Rk + (gR² / 2u² − h) = 0.
2. A launch angle that reaches range R exists only if this quadratic has a real solution, so the discriminant is at least zero: R² − 4 (gR² / 2u²)(gR² / 2u² − h) ≥ 0.
3. Divide by R² and simplify: 1 − g²R² / u⁴ + 2gh / u² ≥ 0, so R² ≤ (u⁴ / g²)(1 + 2gh/u²).
4. The largest possible range is therefore

R_max = (u / g) √(u² + 2gh)

5. At the maximum the discriminant is zero, so the single root is k = R / (2 × gR²/2u²) = u² / (gR_max). Substituting R_max gives

tan θ_opt = u / √(u² + 2gh)

Check: for h = 0, tan θ = 1, so θ = 45° and R_max = u²/g, matching C7. As h increases, the optimum angle falls below 45°.

C9. Impact speed from energy conservation

With no drag, mechanical energy is conserved. Taking ground as zero potential energy, ½mu² + mgh = ½mv², so

v_impact = √(u² + 2gh)

The impact speed depends on u and h only, not on the launch angle and not on the mass. To find the impact angle, note that the horizontal component is still u cos θ. Then v_y = √(v² − u² cos²θ) = √(u² sin²θ + 2gh) (downward), and the angle below the horizontal is tan φ = |v_y| / (u cos θ).

Every example uses g = 9.81 m/s² on Earth unless stated. Intermediate values are kept at full calculator precision and only the final answers are rounded. Final answers are given to three significant figures, which suits data given to two or three significant figures.

Graphs summarise the motion. Remember the general rules: the gradient of a displacement-time graph is velocity, the gradient of a velocity-time graph is acceleration, and the area under a velocity-time graph is displacement. For a projectile launched and landing at the same height with upward positive:

x against t

A straight line through the origin (if x is measured from the launch point). Its gradient is the constant horizontal speed u cos θ. A curved x-t graph would mean the horizontal speed is changing, which signals air resistance.

y against t

An inverted parabola. The gradient at any instant is v_y: positive on the way up, zero at the peak (the flat top of the curve) and negative on the way down. For launch and landing at the same level the curve is symmetrical about the peak time u sin θ / g.

v_x against t

A horizontal straight line at u cos θ. The area under it up to time t equals the horizontal distance x.

v_y against t

A straight line with gradient −g, starting at +u sin θ and crossing zero at the top of the path. The area above the axis is the height gained on the way up; the area below the axis is the distance fallen. For a level-ground launch these two areas are equal in size, so the net displacement is zero.

y against x (the path)

An inverted parabola. This is the shape of the actual trajectory, not a time graph. Each point corresponds to one moment in time, and time is not shown on the axes.

Qualitative effects (A-level)

Air resistance (drag) is a force that opposes the motion through the air and grows as the speed grows. It acts along the direction opposite to the velocity, so it slows the horizontal and vertical components. Compared with the ideal case, a real projectile typically:

• has a shorter range and a lower maximum height;
• has a horizontal velocity that falls continuously (the x-t graph bends over);
• follows an asymmetric path: the descent is steeper than the ascent, and it lands at a steeper angle;
• takes a longer time to fall than to rise, because the speed and so the drag are lower near the top;
• reaches its best range at a launch angle below 45°.

The effect depends strongly on the object. A dense, compact object such as a steel shot barely notices air at modest speeds, while a light, wide object such as a table tennis ball is strongly affected.

The quadratic drag model (extension)

At the speeds of everyday sport, the drag force on a sphere is well modelled by

F_d = ½ ρ C_d A v²

where ρ is the air density (1.225 kg/m³ in the simulation), C_d the drag coefficient (0.47 for a sphere in the simulation), A the cross-sectional area (πd²/4 for a sphere of diameter d) and v the speed. In vector form the drag points opposite to the velocity, so the components of the acceleration are

a_x = −k v v_x    a_y = −g − k v v_y,   k = ρC_dA / (2m)

Notice that both equations now contain the total speed v, so the horizontal and vertical motions are coupled: the independence principle no longer holds exactly. (The forces still act along perpendicular axes, but the size of each drag component depends on both velocity components.)

Terminal velocity

For an object falling vertically, drag grows until it equals the weight, and the acceleration drops to zero. Setting ½ρC_dAv² = mg gives

v_t = √(2mg / (ρC_dA))

With the simulation's values, v_t is 58.2 m/s for the steel shot, 19.6 m/s for the football and 8.56 m/s for the table tennis ball. Compare any launch speed with the terminal speed to judge how important drag will be: when u is much smaller than v_t the ideal model works well; when u is comparable to or larger than v_t it fails badly.

Why the optimum angle drops below 45° (extension)

Drag takes more energy from the flight the longer the projectile stays in the air at high speed. A high-angle shot spends longer in the air and gives drag more time to act, while a lower angle loses less to drag. Balancing these gives a best angle below 45° whose size depends on the ratio of u to v_t. The simulation lets you find it by experiment.

Numerical integration (extension)

Because drag couples the equations, we step forward in small time intervals Δt. In the Euler method, at each step we compute the acceleration from the current velocity, then update: v → v + aΔt and position → position + vΔt. It is easy but its error shrinks only in proportion to Δt, so energy drifts in an unphysical way unless Δt is very small. The fourth-order Runge-Kutta method (RK4) evaluates the acceleration at four carefully weighted points within each step and combines them; its error shrinks in proportion to Δt⁴, so far larger steps give accurate results. A good habit is to halve Δt and check that the answer barely changes.

Research question

A good research question names the variables and the relationship. For example: "How does the launch angle affect the horizontal range of a ball launched at a fixed speed from level ground, and is the relationship consistent with R = u² sin 2θ / g?" This is testable, has a theoretical prediction, and lets you find g from the results.

Variables

Method for a real laboratory

1. Fix a spring launcher or ramp-and-ball system to a bench. Check that it is secure and that the area downrange is clear (see section J).
2. Measure the launch speed independently. Two options: two light gates a known distance apart just after the launch, or a video of the ball against a scale. Repeat several times and take the mean.
3. Set the angle with an inclinometer. Aim for 10° steps from 10° to 80°, plus extra readings near 45°.
4. Use carbon paper over white paper, or a sand tray, to mark where the ball lands. Measure from the launch point to the mark with a tape measure.
5. Take at least three launches at each angle. Discard none without a reason, and note any obvious anomaly.
6. Optionally film the flight at a known frame rate with a scale in view, and use video analysis software to get x-t and y-t data.

Method for the virtual lab

1. Select Earth, a launch height of 0 and drag off. Choose a speed (the default is 20 m/s).
2. Launch at angles from 5° to 85° and record the range each time. Use smaller steps near 45°.
3. Repeat the investigation with drag on, for each of the three projectiles, and compare.
4. Repeat with a different value of g (Moon, Mars, Jupiter) at the same speed and angle, to test the 1/g scaling.
5. Repeat with a launch height of, say, 10 m, to find the best angle and compare with the formula in note C8.

Choosing ranges and repeats

Cover the full useful range of the independent variable, here 0° to 90°, with at least 8 to 10 distinct values so a trend can be seen. Use closer spacing where the dependent variable changes most or where a maximum is expected. In a real lab, a simulation is not a replacement for repeat measurements: the virtual lab is deterministic, so repeating it gives identical results, but real data scatter, and repeats let you estimate the random error.

Types of error

• Random error makes repeated readings scatter unpredictably above and below the true value (for example, the landing spot of a ball varies). Repeating and averaging reduces its effect.
• Systematic error shifts all readings the same way (for example, a protractor zero set 2° wrong, or a tape measure started from the wrong end). Repeating does not remove it; careful calibration and checking against a known value do.
• Resolution is the smallest change an instrument can show. A common estimate of reading uncertainty is half the smallest division for an analogue scale, or the smallest digit for a digital one (check your course's convention).

Uncertainty rules

• Absolute uncertainty has the same units as the quantity, for example ±0.5 m/s.
• Percentage uncertainty = (absolute uncertainty / value) × 100%. Example: u = 20.0 ± 0.5 m/s is a 2.5% uncertainty.
• Adding or subtracting quantities: add the absolute uncertainties.
• Multiplying or dividing: add the percentage uncertainties.
• Raising to a power n: multiply the percentage uncertainty by n. Example: u² has a percentage uncertainty of 2 × 2.5% = 5%.
• Mean of repeats: a simple estimate of the uncertainty is half the range (largest minus smallest), or the standard deviation of the mean if your course uses it.

Using a graph of R against sin 2θ

From R = (u² / g) sin 2θ, a graph of R (vertical axis) against sin 2θ (horizontal axis) is a straight line through the origin with gradient u²/g. This is a better analysis than plotting R against θ, because a straight line is easy to judge by eye. Then

g = u² / gradient

Worked illustration. Suppose u = 20.0 m/s is measured independently, and the best-fit line through the data has gradient 40.8 m. Then g = 400 / 40.8 = 9.80 m/s². To estimate the uncertainty, draw the steepest and the shallowest lines that still pass through all the error bars. Suppose these have gradients 42.0 m and 39.6 m. They give g = 400 / 42.0 = 9.52 m/s² and g = 400 / 39.6 = 10.10 m/s². Half the spread is (10.10 − 9.52) / 2 = 0.29 m/s², so the result is g = 9.8 ± 0.3 m/s². Note the steeper line gives the smaller g, because g varies inversely with the gradient.

Evaluating a conclusion

Compare your result with the accepted value (9.81 m/s² for the simulation's Earth) and ask whether the accepted value lies within your uncertainty range. If it does, your result is consistent with it; if not, look for systematic error. Then comment on: whether the data show a straight line through the origin (a test of the model), how large the scatter is, which measurement contributed the largest percentage uncertainty (usually u, because it is squared), and specific, realistic improvements such as measuring u with light gates, using more repeats, or reducing air resistance by using a denser ball.

Typical resolutions below are indicative. Always use the resolution stated on your own instrument.

Real-laboratory safety

• Wear eye protection whenever anything is launched.
• Make sure the range is clear: nobody downrange, and no windows, equipment or people in the landing zone. Agree a clear signal before each launch.
• Clamp or otherwise secure the launcher so it cannot move or tip when it fires.
• Use soft or light projectiles (for example a foam or plastic ball) rather than heavy objects, unless your teacher or supervisor has approved the setup and carried out a risk assessment.
• Never aim at a person, never look into the launcher, and retrieve projectiles only when everyone has agreed it is safe.
• Follow your school or institution's risk assessment and your teacher's instructions. These notes are general guidance and do not replace them.

What the simulation idealises

The simulation is a model, not reality. Use it to explore ideas and to practise analysis, and always judge whether its simplifying assumptions are reasonable for a given real situation.

Command words

• State or Give: a short answer, no explanation needed.
• Calculate or Determine: a numerical answer with working and a unit.
• Show that: you are given the final answer, so each step must be visible and the answer must be quoted to one more significant figure than the given value.
• Explain: give a reason using physics ideas, linking cause to effect.
• Describe: say what happens, without the reason.
• Sketch: show the shape and key features, with labelled axes; precise values are not needed.
• Derive: start from a stated principle and reach the result in logical steps.
• Estimate: use sensible approximate values and say what they are.

Command words and their exact meanings differ slightly between exam boards, so check the glossary published by your own board.

Mark scheme patterns

Mark schemes typically award separate marks for a correct method (for example, resolving the velocity), correct substitution, a correct answer, and a correct unit. A wrong answer with clear working can still earn most of the marks, while a correct answer with no working may earn fewer than full marks on a multi-mark question. Write your working in a logical order so that an examiner can follow it.

Good habits

• Significant figures: give the final answer to the same number of significant figures as the least precise data value (usually 2 or 3). Do not round intermediate steps.
• Units: include them on every answer. Check that they make sense (a launch speed of 400 m/s is not plausible for a thrown ball).
• Sign convention: state it at the start, for example "upward is positive, so a = −9.81 m/s²".
• Sketch graphs: label the axes with quantity and unit, mark where the curve crosses zero, and show whether a line is straight or curved.
• Check plausibility: does the range look right, is a time positive, does a height exceed the maximum?

Explaining independence in words

A strong answer is: "The only force acting is the weight, which is vertical. So there is no horizontal acceleration and the horizontal velocity stays constant. The vertical velocity changes at 9.81 m/s every second. The two motions share the same time of flight, but one does not affect the other."

Typical mark-allocation patterns (illustrative)

• 1 mark: a single recall or one-step calculation, for example "State the horizontal acceleration of a projectile" (zero).
• 2 marks: a short explanation or two-step calculation, for example find the vertical component, then the time to the top.
• 4 marks: a multi-step calculation with a unit, for example horizontal launch from a height, finding time and range.
• 6 marks: an extended response or a longer problem with explanation, such as comparing two launch angles, or explaining the effects of air resistance with a sketch. Use a clear structure: state the principle, apply it, conclude.

This map is a guide to commonly taught topic areas. It is not an official statement from any exam board. Always check your current specification.

This section is extension material for HL, AP Physics C and university courses. It is not needed for standard A-level or AP Physics 1 examinations.

The safety parabola (envelope of trajectories)

Fix the launch speed u and the launch point, but allow any angle. Which points in the plane can be reached? Use the trajectory equation from the origin, y = x tan θ − gx²(1 + tan²θ) / (2u²), and treat it as a quadratic in k = tan θ:

(gx² / 2u²) k² − xk + (y + gx² / 2u²) = 0

A point (x, y) is reachable only if this has a real solution, so the discriminant is not negative. This gives the boundary

y = u² / (2g) − gx² / (2u²)

called the safety parabola. Points under it can be reached by some launch angle; points above it cannot. It meets the ground at x = u²/g (the maximum level range) and has peak height u²/(2g) (the maximum height, for a vertical launch). Every individual trajectory touches this envelope at exactly one point.

Optimum angle with drag

With quadratic drag there is no neat formula. The best angle depends on the dimensionless ratio of the launch speed to the terminal speed, u/v_t. When this ratio is small the optimum is near 45°, and as it grows the optimum decreases, with larger drag making the loss at high angles worse. In the simulation, try the table tennis ball at 60 m/s and compare it with the steel shot at the same speed.

Projectile on an incline

Launch at angle θ above the horizontal from the foot of a plane rising at angle β to the horizontal. Resolving parallel and perpendicular to the slope, and noting that gravity then has components along both axes, the range measured along the slope is

R = 2u² cos θ sin(θ − β) / (g cos²β)

The range is a maximum when θ = 45° + β/2, and then R_max = u² / (g(1 + sin β)). For β = 0 this reduces to u²/g, as it should. A downhill slope is handled by making β negative.

Rotating Earth

Because the Earth rotates, a frame fixed to the ground is not an exact inertial frame. For long-range projectiles (artillery shells, ballistic missiles) the Coriolis effect produces a small sideways deflection that grows with flight time. For a sport or a classroom launch the effect is far smaller than other sources of error and can be ignored.

Magnus effect (qualitative)

A spinning ball drags a thin layer of air around with it. The air flows faster past one side than the other, which produces a net force perpendicular to both the velocity and the spin axis. Topspin pushes a ball downward so that it dips sooner; backspin lifts it and extends the flight; sidespin curves it sideways (a "swerve" in football). The size of the force depends on speed, spin rate and surface. The simulation does not include it.

Lagrangian and Hamiltonian formulations (mention only)

In advanced mechanics the same problem is posed through energy. The Lagrangian is L = ½m(ẋ² + ẏ²) − mgy, and the Euler-Lagrange equations return ẍ = 0 and ÿ = −g, exactly the equations used above. The Hamiltonian is the total energy expressed in terms of momentum and position. The method becomes useful when constraints or complicated forces are present.

1. A projectile moves under gravity alone after launch (drag ignored in the ideal model).
2. Horizontal and vertical motions are independent but share the same time.
3. Horizontal acceleration is zero, so the horizontal velocity is constant.
4. Vertical acceleration is −g (9.81 m/s² on Earth) everywhere on the path, including the top.
5. Resolve the launch velocity: u_x = u cos θ, u_y = u sin θ.
6. Choose a positive direction and keep to it throughout.
7. At the top, v_y = 0 but the speed is u cos θ, not zero.
8. SUVAT applies to each component separately, not along the curved path.
9. Time of flight from a height: T = [u sin θ + √(u² sin²θ + 2gh)] / g.
10. On level ground, T = 2u sin θ / g.
11. Maximum height gained is u² sin²θ / (2g).
12. Level-ground range: R = u² sin 2θ / g; use it only when launch and landing heights match.
13. Complementary angles (θ and 90° − θ) give the same level-ground range.
14. 45° gives the greatest level-ground range, u²/g.
15. From a height, the best angle is tan θ = u / √(u² + 2gh), below 45°.
16. Impact speed without drag is √(u² + 2gh), independent of angle and mass.
17. The path is a parabola; x-t is a straight line, y-t a parabola, v_x-t flat, and v_y-t a straight line of gradient −g.
18. Range, height and flight time are all proportional to 1/g for fixed launch conditions.
19. Drag shortens range, makes the path asymmetric, and lowers the optimum angle; terminal speed is √(2mg/(ρC_dA)).
20. A graph of R against sin 2θ is a straight line through the origin of gradient u²/g, so g = u²/gradient.

Acceleration

The rate of change of velocity with time; a vector.

Air resistance (drag)

A force opposing motion through air, increasing with speed.

Angle of projection

The angle between the launch velocity and the horizontal.

Complementary angles

Two angles that add up to 90°.

Component

The part of a vector along a chosen direction, usually horizontal or vertical.

Constant acceleration

Acceleration whose size and direction do not change, so SUVAT applies.

Drag coefficient (C_d)

A dimensionless number describing how much drag a shape produces at a given speed.

Envelope

The curve that bounds all possible trajectories for a fixed launch speed; also called the safety parabola.

Euler method

A simple numerical method that advances a solution in small time steps using the current rate of change.

Free fall

Motion under gravity alone.

Gradient

The slope of a graph, the change in the vertical quantity divided by the change in the horizontal quantity.

Gravitational field strength (g)

The force per unit mass due to gravity; equals the free-fall acceleration in a vacuum.

Independence of motion

The principle that horizontal and vertical motions of a projectile do not affect each other (ideal case).

Inertia

The tendency of an object to keep its velocity unless a net force acts.

Magnus effect

A sideways force on a spinning object moving through a fluid.

Maximum height

The greatest height reached above the reference level.

Parabola

The curve described by a quadratic equation; the ideal projectile path.

Projectile

An object in flight acted on by gravity (and possibly drag) only.

Quadratic drag

A drag force proportional to the square of the speed.

Random error

Unpredictable scatter in repeated measurements.

Range

The horizontal distance from the launch point to the landing point.

Resolution

The smallest change an instrument can detect.

Resolving a vector

Splitting a vector into perpendicular components.

Runge-Kutta (RK4)

A higher-accuracy numerical method that combines four estimates of the rate of change in each step.

Scalar

A quantity with size only, such as speed or time.

Speed

The size of the velocity; a scalar.

SUVAT

The set of equations for constant acceleration using s, u, v, a and t.

Systematic error

A consistent offset in measurements in one direction.

Terminal velocity

The constant speed reached when drag balances weight.

Time of flight

The time from launch to landing.

Trajectory

The path followed by a projectile.

Uncertainty

The range of values within which the true value is expected to lie.

Vector

A quantity with both size and direction, such as velocity or displacement.

Velocity

The rate of change of displacement; a vector with speed and direction.

Weight

The gravitational force on an object, mg.

Only titles and publishers are given, together with stable home pages. Search each site for the title to reach the exact page.

• OpenStax, University Physics Volume 1, Chapter 4 "Motion in Two and Three Dimensions", section 4.3 "Projectile Motion". Free and open textbook. https://openstax.org/
• OpenStax, College Physics. Free algebra-based textbook with a chapter on two-dimensional kinematics, including projectile motion. https://openstax.org/
• Bureau International des Poids et Mesures (BIPM), The International System of Units (SI Brochure), which records the conventional standard value of gravity as 9.80665 m/s². https://www.bipm.org/
• NASA, Planetary Fact Sheets (NASA Space Science Data Coordinated Archive), for the surface gravity of the Moon, Mars and Jupiter. Values used in the simulation: Moon 1.62, Mars 3.71, Jupiter 24.79 m/s². https://nssdc.gsfc.nasa.gov/planetary/factsheet/
• Taylor, J. R., Classical Mechanics, University Science Books. Contains a full treatment of projectiles with linear and quadratic drag (extension reading).

The simulation uses g = 9.81 m/s² for Earth, a rounded value near the conventional 9.80665 m/s².