–Select–
Initial Velocity (v₀)
Final Velocity (v)
Acceleration (a)
Time (t)
Displacement (Δx)
–Select–
Initial Velocity (v₀)
Final Velocity (v)
Acceleration (a)
Time (t)
Displacement (Δx)
–Select–
Initial Velocity (v₀)
Final Velocity (v)
Acceleration (a)
Time (t)
Displacement (Δx)
–Select–
Initial Velocity (v₀)
Final Velocity (v)
Acceleration (a)
Time (t)
Displacement (Δx)
Result
Enter values and select variables to calculate.
Understanding Kinematic Equations
Kinematic equations are a set of formulas used in physics to describe the motion of objects. They relate fundamental quantities of motion, such as displacement, velocity, acceleration, and time, under the assumption of constant acceleration. These equations are foundational for analyzing projectile motion, linear motion, and many other scenarios in classical mechanics.
The Five Standard Kinematic Equations (for constant acceleration):
Equation 1 (Velocity-Time):v = v₀ + at
Relates final velocity (v), initial velocity (v₀), acceleration (a), and time (t).
Equation 2 (Displacement-Time):Δx = v₀t + ½at²
Relates displacement (Δx), initial velocity (v₀), time (t), and acceleration (a).
Equation 3 (Displacement-Velocity):v² = v₀² + 2aΔx
Relates final velocity (v), initial velocity (v₀), acceleration (a), and displacement (Δx). This equation is useful when time is unknown or not needed.
Equation 4 (Displacement-Average Velocity):Δx = ½(v₀ + v)t
Relates displacement (Δx), initial velocity (v₀), final velocity (v), and time (t). This is derived from the definition of average velocity.
Equation 5 (Displacement-Average Velocity with Constant Acceleration):Δx = vt - ½at²
Relates displacement (Δx), final velocity (v), acceleration (a), and time (t).
How to Use This Calculator:
This calculator allows you to solve for one unknown kinematic variable when three other variables are known. Follow these steps:
Select the first known variable and enter its value.
Select the second known variable and enter its value.
Select the third known variable and enter its value.
Choose the variable you wish to calculate from the "Variable to Calculate" dropdown.
Click "Calculate".
The calculator will determine which kinematic equation is most suitable based on the known and unknown variables and provide the calculated result with appropriate units. Ensure your inputs are in a consistent set of units (e.g., meters per second for velocity, seconds for time, meters per second squared for acceleration, meters for displacement).
Example:
Suppose a car starts from rest (v₀ = 0 m/s) and accelerates uniformly at 2 m/s² for 5 seconds. What is its final velocity?
Known 1: Initial Velocity (v₀) = 0
Known 2: Acceleration (a) = 2
Known 3: Time (t) = 5
To Calculate: Final Velocity (v)
The calculator will use the equation v = v₀ + at. With these inputs, the result would be 10 m/s.
function getElementValue(id) {
var element = document.getElementById(id);
if (element && element.value !== "") {
var value = parseFloat(element.value);
return isNaN(value) ? null : value;
}
return null;
}
function getSelectedText(id) {
var selectElement = document.getElementById(id);
if (selectElement && selectElement.value) {
return selectElement.options[selectElement.selectedIndex].text;
}
return null;
}
function setResults(value, unit, variableName) {
var resultDiv = document.getElementById("result");
var unitsDiv = document.getElementById("units");
if (value === null) {
resultDiv.textContent = "Error: Calculation failed.";
unitsDiv.textContent = "";
} else {
resultDiv.textContent = value.toFixed(4); // Display with 4 decimal places
unitsDiv.textContent = unit;
}
}
function updateAvailableVariables() {
var known1 = document.getElementById("known1").value;
var known2 = document.getElementById("known2").value;
var known3 = document.getElementById("known3").value;
var toCalculate = document.getElementById("toCalculate");
var options = toCalculate.options;
for (var i = 0; i < options.length; i++) {
var optionValue = options[i].value;
if (optionValue === "") continue; // Skip the default option
var isAlreadyKnown = (optionValue === known1 || optionValue === known2 || optionValue === known3);
if (isAlreadyKnown) {
options[i].disabled = true;
} else {
options[i].disabled = false;
}
}
}
function calculateKinematic() {
var known1Type = document.getElementById("known1").value;
var val1 = getElementValue("val1");
var known2Type = document.getElementById("known2").value;
var val2 = getElementValue("val2");
var known3Type = document.getElementById("known3").value;
var val3 = getElementValue("val3");
var calcType = document.getElementById("toCalculate").value;
var v0 = null, v = null, a = null, t = null, dx = null;
var units = { v0: "m/s", v: "m/s", a: "m/s²", t: "s", dx: "m" };
var calculatedValue = null;
var calculatedUnit = "";
// Assign input values to variables
if (known1Type === "initial_velocity") { v0 = val1; }
else if (known1Type === "final_velocity") { v = val1; }
else if (known1Type === "acceleration") { a = val1; }
else if (known1Type === "time") { t = val1; }
else if (known1Type === "displacement") { dx = val1; }
if (known2Type === "initial_velocity") { v0 = val2; }
else if (known2Type === "final_velocity") { v = val2; }
else if (known2Type === "acceleration") { a = val2; }
else if (known2Type === "time") { t = val2; }
else if (known2Type === "displacement") { dx = val2; }
if (known3Type === "initial_velocity") { v0 = val3; }
else if (known3Type === "final_velocity") { v = val3; }
else if (known3Type === "acceleration") { a = val3; }
else if (known3Type === "time") { t = val3; }
else if (known3Type === "displacement") { dx = val3; }
// Basic validation for number of inputs
var inputCount = 0;
if (v0 !== null) inputCount++;
if (v !== null) inputCount++;
if (a !== null) inputCount++;
if (t !== null) inputCount++;
if (dx !== null) inputCount++;
if (inputCount type !== "");
if (new Set(types).size !== types.length) {
setResults(null, "", "");
document.getElementById("result").textContent = "Duplicate input variables selected. Please select unique variables.";
return;
}
// Calculations based on the variable to calculate
if (calcType === "initial_velocity") {
calculatedUnit = units.v0;
if (v !== null && a !== null && t !== null) { // v = v₀ + at => v₀ = v – at
calculatedValue = v – (a * t);
} else if (v !== null && a !== null && dx !== null) { // v² = v₀² + 2aΔx => v₀² = v² – 2aΔx => v₀ = sqrt(v² – 2aΔx)
if (v*v – 2*a*dx >= 0) {
calculatedValue = Math.sqrt(v*v – 2*a*dx);
}
} else if (v !== null && t !== null && dx !== null) { // Δx = ½(v₀ + v)t => 2Δx/t = v₀ + v => v₀ = (2Δx/t) – v
if (t !== 0) {
calculatedValue = (2 * dx / t) – v;
}
} else if (v0 !== null && a !== null && t !== null && dx === null) { // Use provided v0 if it's one of the inputs
// This case handles when v0 is provided and we are asked to calculate v0 again. It's redundant but safe.
// If v0 is provided, it should be the result, unless it's a contradiction.
// For this calculator's logic, we assume the user wants to derive it if possible from other inputs.
// However, if v0 is explicitly provided as an input, and we are asked to calculate v0,
// we should ensure consistency or signal an issue.
// For simplicity, let's assume if v0 is given, it's not what we are calculating *unless* it's needed to resolve other unknowns.
// If calcType is v0 and v0 is provided, and we CANNOT solve it from other 3 inputs, it implies an issue.
// If we CAN solve it from other 3 inputs, we should use that derived value.
// Let's prioritize derived values if possible. If not, and v0 is provided, we can just confirm it.
// But if we have v0, a, t provided, and want to calculate v0 – this implies a potential error in input selection.
// The current logic covers cases where v0 is NOT provided.
// If v0 IS provided, it means one of the other inputs (v, a, t, dx) is NOT provided.
// Example: Need to calculate v0. Inputs: v, a, t. Correct.
// Example: Need to calculate v0. Inputs: v0, a, t. Ambiguous.
// This code prioritizes solving for the requested calcType using the *other* three provided variables.
// If the provided v0 is meant to be the answer, then it should not be one of the 3 inputs.
} else if (v0 !== null && t !== null && dx !== null) { // Δx = v₀t + ½at² -> a = (2*(Δx – v₀t))/t²
// This equation is not for finding v0
}
} else if (calcType === "final_velocity") {
calculatedUnit = units.v;
if (v0 !== null && a !== null && t !== null) { // v = v₀ + at
calculatedValue = v0 + (a * t);
} else if (v0 !== null && a !== null && dx !== null) { // v² = v₀² + 2aΔx => v = sqrt(v₀² + 2aΔx)
calculatedValue = Math.sqrt(v0*v0 + 2*a*dx);
} else if (v0 !== null && t !== null && dx !== null) { // Δx = ½(v₀ + v)t => 2Δx/t = v₀ + v => v = (2Δx/t) – v₀
if (t !== 0) {
calculatedValue = (2 * dx / t) – v0;
}
} else if (v !== null && t !== null && dx !== null) { // Δx = vt – ½at² => a = (2*(vt – Δx))/t²
// This equation is not for finding v
}
} else if (calcType === "acceleration") {
calculatedUnit = units.a;
if (v !== null && v0 !== null && t !== null) { // v = v₀ + at => a = (v – v₀)/t
if (t !== 0) {
calculatedValue = (v – v0) / t;
}
} else if (v !== null && v0 !== null && dx !== null) { // v² = v₀² + 2aΔx => a = (v² – v₀²)/(2Δx)
if (dx !== 0) {
calculatedValue = (v*v – v0*v0) / (2 * dx);
}
} else if (v !== null && v0 !== null && t !== null && dx !== null) { // Δx = v₀t + ½at² => a = (2*(Δx – v₀t))/t²
if (t !== 0) {
calculatedValue = (2 * (dx – v0 * t)) / (t * t);
}
} else if (v !== null && t !== null && dx !== null) { // Δx = vt – ½at² => a = (2*(vt – Δx))/t²
if (t !== 0) {
calculatedValue = (2 * (v * t – dx)) / (t * t);
}
}
} else if (calcType === "time") {
calculatedUnit = units.t;
if (v !== null && v0 !== null && a !== null) { // v = v₀ + at => t = (v – v₀)/a
if (a !== 0) {
calculatedValue = (v – v0) / a;
}
} else if (v !== null && v0 !== null && dx !== null && a !== null) { // Quadratic equation for t from Δx = v₀t + ½at²
// at² + 2v₀t – 2Δx = 0
// Using quadratic formula: t = [-b ± sqrt(b² – 4ac)] / 2a
// Here: A=a, B=2v₀, C=-2Δx
var A = a;
var B = 2 * v0;
var C = -2 * dx;
var discriminant = B*B – 4*A*C;
if (discriminant >= 0) {
var t1 = (-B + Math.sqrt(discriminant)) / (2 * A);
var t2 = (-B – Math.sqrt(discriminant)) / (2 * A);
// Time must be positive in most physical contexts
if (t1 > 0) calculatedValue = t1;
if (t2 > 0 && t2 t = 2Δx / (v₀ + v)
if (v0 + v !== 0) {
calculatedValue = (2 * dx) / (v0 + v);
}
} else if (v !== null && dx !== null && a !== null) { // Quadratic equation for t from Δx = vt – ½at²
// at² – 2vt + 2Δx = 0
// Using quadratic formula: t = [-b ± sqrt(b² – 4ac)] / 2a
// Here: A=a, B=-2v, C=2Δx
var A = a;
var B = -2 * v;
var C = 2 * dx;
var discriminant = B*B – 4*A*C;
if (discriminant >= 0) {
var t1 = (-B + Math.sqrt(discriminant)) / (2 * A);
var t2 = (-B – Math.sqrt(discriminant)) / (2 * A);
// Time must be positive in most physical contexts
if (t1 > 0) calculatedValue = t1;
if (t2 > 0 && t2 Δx = (v² – v₀²)/(2a)
if (a !== 0) {
calculatedValue = (v*v – v0*v0) / (2 * a);
}
} else if (v !== null && t !== null && a !== null) { // Δx = vt – ½at²
calculatedValue = (v * t) – (0.5 * a * t * t);
}
}
setResults(calculatedValue, calculatedUnit, calcType);
}
// Initial call to set up dropdowns correctly on load
document.addEventListener('DOMContentLoaded', updateAvailableVariables);