Calculate the expected trajectory and landing point of an object on Mars.
m/s
degrees
m/s²
kg/m³
dimensionless
m²
kg
Understanding the Moore-Marsden Calculator
The Moore-Marsden calculator is a specialized tool designed to estimate the trajectory of an object undergoing projectile motion under the specific gravitational and atmospheric conditions of Mars. Unlike Earth-based calculations, this model accounts for Mars' lower gravity and its thin atmosphere, which significantly affect how an object flies through space.
Key Concepts and Formulas:
This calculator simplifies the complex physics of atmospheric entry and projectile motion to provide an estimated range and peak altitude. It typically considers the following factors:
Initial Velocity (v₀): The speed at which the object is launched. Higher velocity generally leads to a longer range.
Launch Angle (θ): The angle relative to the horizontal at which the object is launched. For ideal projectile motion without air resistance, 45 degrees yields the maximum range. However, air resistance modifies this optimal angle.
Mars Surface Gravity (g): Mars has approximately 38% of Earth's gravity (about 3.71 m/s²). This lower gravity means an object will stay airborne longer and travel further horizontally, assuming all other factors are equal.
Mars Air Density (ρ): Mars has a very thin atmosphere, with density roughly 1-2% of Earth's at sea level (around 0.020 kg/m³). This significantly reduces air resistance compared to Earth.
Drag Coefficient (Cd): A dimensionless quantity that describes how effectively an object cuts through the air. It depends on the object's shape.
Cross-sectional Area (A): The area of the object facing the direction of motion. A larger area leads to greater drag.
Object Mass (m): The mass of the object. A heavier object is less affected by air resistance and drag.
The Physics at Play (Simplified):
The calculator models the forces acting on the projectile:
Gravity: Pulling the object downwards (Force = m * g).
Drag Force: Resisting the object's motion through the Martian atmosphere. The drag force (Fd) is approximated by:
Fd = 0.5 * ρ * v² * Cd * A where ρ is air density, v is velocity, Cd is the drag coefficient, and A is the cross-sectional area.
The calculator solves differential equations that describe the x and y components of motion, incorporating both gravity and drag. Due to the complexity, many online calculators use numerical methods or simplified analytical solutions that make approximations.
Use Cases:
Mission Planning: Estimating the landing trajectory of probes, rovers, or descent stages.
Scientific Research: Modeling atmospheric re-entry phenomena for Martian conditions.
Educational Tools: Demonstrating the principles of projectile motion and atmospheric drag in a different planetary environment.
Future Colonization Concepts: Hypothetically calculating the trajectory of objects launched or dropped on Mars.
This calculator provides an approximation. Actual trajectories can be influenced by factors such as wind, atmospheric variations, object spin, and the precise shape of the object, which are often simplified or excluded in basic models.
function calculateMooreMarsden() {
var v0 = parseFloat(document.getElementById("initialVelocity").value);
var angleRad = parseFloat(document.getElementById("launchAngle").value) * Math.PI / 180;
var g = parseFloat(document.getElementById("marsGravity").value);
var rho = parseFloat(document.getElementById("airDensity").value);
var Cd = parseFloat(document.getElementById("dragCoefficient").value);
var A = parseFloat(document.getElementById("objectArea").value);
var m = parseFloat(document.getElementById("objectMass").value);
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = "; // Clear previous results
// Input validation
if (isNaN(v0) || isNaN(angleRad) || isNaN(g) || isNaN(rho) || isNaN(Cd) || isNaN(A) || isNaN(m) ||
v0 < 0 || g <= 0 || rho < 0 || Cd < 0 || A <= 0 || m <= 0) {
resultDiv.innerHTML = "Please enter valid positive numbers for all fields.";
return;
}
if (document.getElementById("launchAngle").value 180) {
resultDiv.innerHTML = "Launch angle must be between 0 and 180 degrees.";
return;
}
// — Simplified Moore-Marsden Calculation —
// This is a simplified model and may not perfectly represent complex simulations.
// A common approach involves numerical integration (e.g., Euler method or Runge-Kutta).
// For a direct analytical solution with drag, it becomes extremely complex.
// We will use a common approximation focusing on the dominant factors.
// Calculate initial horizontal and vertical velocities
var v0x = v0 * Math.cos(angleRad);
var v0y = v0 * Math.sin(angleRad);
// — Approximation 1: Ignoring Drag for a Baseline —
var timeToPeakApprox = v0y / g;
var peakHeightApprox = (v0y * v0y) / (2 * g);
var totalFlightTimeApprox = 2 * v0y / g;
var rangeApprox = v0x * totalFlightTimeApprox;
// — Approximation 2: Incorporating Drag (Simplified Drag Force) —
// This is a highly simplified drag model. Real simulations use iterative methods.
// We'll estimate the effect of drag on range and peak height.
// A common simplification is to use an effective gravity or to iterate.
// Let's use a basic iterative approach for a better estimate.
var dt = 0.01; // Time step for simulation
var x = 0;
var y = 0;
var vx = v0x;
var vy = v0y;
var t = 0;
var max_y = 0;
var final_x = 0;
var final_t = 0;
// Calculate the drag factor to simplify the iterative calculation
// This factor combines Cd, rho, A, and m.
var dragFactor = 0.5 * rho * Cd * A / m;
while (y >= 0) {
// Update max height
if (y > max_y) {
max_y = y;
}
// Calculate current speed
var v = Math.sqrt(vx * vx + vy * vy);
// Calculate drag force components (opposite to velocity vector)
var Fdx = -dragFactor * v * vx;
var Fdy = -dragFactor * v * vy;
// Calculate acceleration components (gravity + drag)
var ax = Fdx;
var ay = -g + Fdy;
// Update velocities using Euler method
vx += ax * dt;
vy += ay * dt;
// Update positions
x += vx * dt;
y += vy * dt;
// Update time
t += dt;
// Store last position before hitting ground
if (y < 0) {
final_x = x;
final_t = t;
break; // Exit loop when object hits ground (or goes below)
}
}
// Prevent NaN if the loop doesn't execute as expected (e.g., immediate downward launch)
if (isNaN(final_x) || !isFinite(final_x)) final_x = 0;
if (isNaN(max_y) || !isFinite(max_y)) max_y = 0;
if (isNaN(final_t) || !isFinite(final_t)) final_t = 0;
resultDiv.innerHTML = "Estimated Range: " + final_x.toFixed(2) + " m Peak Altitude: " + max_y.toFixed(2) + " m Flight Time: " + final_t.toFixed(2) + " s";
}