Linear Motion Calculator

MRU vs. MRUV — What's the Difference?

MRU (uniform rectilinear motion) describes an object moving at constant velocity — no speeding up, no slowing down. Its position-vs-time graph is a straight line, and its velocity-vs-time graph is a flat horizontal line. MRUV (uniformly accelerated rectilinear motion) adds a constant acceleration: velocity changes linearly with time, and position traces out a parabola instead of a straight line.

Real-world motion is rarely purely one or the other over a long period, but plenty of short intervals approximate each well: a car cruising on a flat highway at a fixed speed behaves like MRU, while the same car accelerating from a stoplight behaves like MRUV until it settles into a new cruising speed. Recognizing which model fits a given interval is the first step in solving almost any kinematics problem.

The Core Equations

$$x(t) = x_0 + v_0 t + \frac{1}{2} a t^2 \qquad v(t) = v_0 + a t$$

With $a = 0$, the position equation reduces to plain MRU. If acceleration opposes the direction of motion, the object decelerates — set $a < 0$ for an object moving in the positive direction to see it slow down and find its stopping time at $t = -v_0/a$.

The Third Kinematic Equation — No Time Required

When you don't know (or don't need) the time, this equation connects velocity, acceleration, and displacement directly:

$$v^2 = v_0^2 + 2a(x - x_0)$$

It's derived by eliminating $t$ from the two core equations above, and it's especially useful for questions like "how fast is it going after traveling this distance?" without solving for time as an intermediate step.

Solving Meeting-Point Problems

The classic "two cars start at different points and speeds — when do they meet?" problem is just setting both position equations equal and solving for t. If both move at constant velocity, it's a simple linear equation. If either one accelerates, it becomes a quadratic with up to two mathematically valid times — only the earliest one within your simulated window is usually the physically meaningful answer.

Worked Example: Two Cars Closing In

Car A starts at position 0 moving at a constant 20 m/s. Car B starts 500 m ahead, moving toward Car A at a constant 10 m/s (so its velocity is −10 m/s in Car A's direction). Setting the two position equations equal: $20t = 500 - 10t$, so $30t = 500$, giving $t \approx 16.67$ s. Plugging that back into either equation gives a meeting point of $20 \times 16.67 \approx 333.3$ m from Car A's start — a good way to sanity-check the calculator's own meeting-point solver with round numbers.

Reading Position, Velocity, and Acceleration Graphs Together

The three graphs this calculator draws are mathematically linked: velocity is the slope of the position graph at each instant, and acceleration is the slope of the velocity graph. A straight-line position graph means zero slope-of-slope, i.e., zero acceleration (MRU). A curving, parabolic position graph means the velocity graph is a sloped straight line — constant acceleration (MRUV). Learning to read the shape of one graph and predict the shape of the other two is one of the most useful intuitions in introductory kinematics.

A Brief History of Kinematics

The formal study of motion without regard to its causes — what's now called kinematics — traces its decisive breakthrough to Galileo Galilei's work in the early 1600s, where he established through careful experimentation that objects under constant acceleration cover distance in proportion to the square of elapsed time, the same relationship encoded in the kinematic equations used here. Galileo's approach was revolutionary because it separated the description of motion (kinematics) from its cause (what we now call dynamics), letting him make precise, testable mathematical predictions about falling and rolling bodies well before Newton later explained gravity's role in causing that motion. Newton's three laws of motion, published in 1687, provided the causal framework that connected force and acceleration, but the purely descriptive kinematic equations — position, velocity, and acceleration as functions of time — remain foundational to physics education essentially unchanged from their early form.

Common Kinematics Mistakes

Mixing up average velocity and instantaneous velocity is one of the most frequent errors — the equations here calculate instantaneous quantities at a specific time, not an average over an interval, and the two only coincide under constant acceleration. Sign errors are another persistent issue: velocity and acceleration are vector quantities, and forgetting that "negative" values represent direction (not simply "slowing down") leads to mistakes when an object is decelerating versus moving backward. Forgetting to check which of the four kinematic variables (time, initial velocity, acceleration, displacement) is unknown before picking a formula can also send you looking for a value the wrong equation doesn't provide.

Kinematics Terms You Should Know

Displacement — the straight-line change in position from start to end, distinct from total distance traveled, which can be larger if the path isn't straight.

Velocity — the rate of change of displacement over time, a vector quantity that includes direction, unlike speed.

Acceleration — the rate of change of velocity over time; constant acceleration is the defining condition of MRUV (uniformly accelerated motion).

MRU vs. MRUV — MRU (uniform rectilinear motion) describes constant-velocity, zero-acceleration motion; MRUV (uniformly accelerated rectilinear motion) describes motion under constant, nonzero acceleration.