Calculator Suite

Linear Motion Calculator

Calculate final velocity, initial velocity, acceleration, or time using v = u + at

Linear Motion Problem Setup

Use v = u + at to solve for final velocity, initial velocity, acceleration, or time

Common Scenarios

Car Acceleration

Car accelerating from rest at 3 m/s² for 10 seconds

Vehicle Braking

How long to stop from highway speed

Free Fall

Object falling under gravity for 3 seconds

Quick Presets

Earth Gravity (-9.81 m/s²)

The 4 Kinematic Equations

Essential formulas for motion with constant acceleration

1. Velocity-Time

v = u + at

Use when you don't know displacement ( $s$ ).

2. Position-Time

s = ut + \frac{1}{2}at^2

Use when you don't know final velocity ( $v$ ).

3. Velocity-Displacement

v^2 = u^2 + 2as

Use when you don't know time ( $t$ ).

4. Average Velocity

s = \frac{(u + v)}{2}t

Use when you don't know acceleration ( $a$ ).

Understanding Linear Motion Calculator

TL;DR

Linear motion describes objects moving in a straight line with constant acceleration. Use these calculators to solve for velocity ( $v, u$ ), time ( $t$ ), acceleration ( $a$ ), or displacement ( $s$ ).

What is Linear Motion?

Linear motion (or rectilinear motion) implies movement along a single dimension. In physics, we usually analyze this under the condition of Uniformly Accelerated Motion (UAM), which means the rate of change of velocity is constant.

How to Use This Calculator

Select your Goal: Choose the variable you want to solve for (e.g., "Final Velocity").
Identify Inputs: Enter the values you know from your problem statement.
Check Directions: Assign positive (+) for one direction (e.g., Up/Right) and negative (-) for the opposite (e.g., Down/Left).
Calculate: The tool will perform the algebra and show the steps.

Real-World Example: Highway Acceleration

Scenario:

A sports car accelerates from a dead stop to highway speed (27 m/s, approx 60 mph) in 5 seconds. What is the acceleration?

Calculation:

Given: $u = 0$ , $v = 27$ , $t = 5$
Find: $a$
Formula: $v = u + at \Rightarrow a = \frac{v-u}{t}$
Result: $a = \frac{27-0}{5} = \textbf{5.4 m/s}^2$

3 Key Checks (Why It Matters)

Sign Convention

Gravity is usually negative ( $-9.8$ ). Braking opposes motion (opposite sign).

Hidden Variables

"At rest" means $v=0$ or $u=0$ . "Dropped" implies $u=0$ .

Common Pitfalls

Mixing units (km/h with seconds)
Forgetting that $g$ is negative
Confusing scalar speed with vector velocity

Unit Consistency

Don't mix minutes with seconds or km/h with m/s. Convert everything to SI first.

Assumptions & Limitations

Constant Acceleration: These formulas fail if acceleration changes (jerk).
Negligible Air Resistance: We assume a vacuum. In reality, drag reduces speed.
Point Particle: We ignore the rotation or size of the object.

Video Tutorials

Intro to Motion (Khan Academy)

Kinematic Equations Explained

Frequently Asked Questions

When can I use these formulas?

Only when acceleration is constant. If acceleration changes (like a rocket engine throttle changing), you must use calculus (derivatives/integrals).

Does object mass affect the answer?

No. In pure kinematics, mass is not a variable. Galileo proved that (ignoring air resistance) heavy and light objects fall at the exact same rate.

What is the difference between speed and velocity?

Speed is a scalar (magnitude only, e.g., "50 mph"). Velocity is a vector (magnitude + direction, e.g., "50 mph North"). This calculator works with vector velocity.

What is "deceleration"?

Deceleration is just negative acceleration (acceleration in the opposite direction of motion). If you are moving right (+), braking is acceleration left (-).