Kinematics is the branch of physics that describes how things move, without asking why they move that way. Dynamics, covered through force, momentum, and impulse below, explains the causes behind that motion. Together they form the foundation for almost everything else in physics. Engineering a car's braking system, calculating a rocket's trajectory, and understanding why a seatbelt saves lives in a crash all trace back to the same handful of core relationships between position, velocity, acceleration, and time.
This guide walks through those relationships in a logical build-up, starting with the basic distinction between speed and velocity, moving into acceleration and the forces that cause it, then momentum and impulse (the physics behind collisions), and finishing with two of the most common applied scenarios: projectile motion and free fall. Each step includes a worked example and a link to the calculator that automates the formula.
Step 1: Speed vs Velocity, What's the Difference
Speed is how fast something is moving, measured as distance traveled divided by time elapsed, a scalar quantity with magnitude only. Velocity adds direction to that picture, measured as displacement (straight-line distance from start to end point, including direction) divided by time. The distinction matters most whenever a path isn't a straight line. A runner completing a 400-meter lap in 60 seconds has an average speed of about 6.67 m/s, but an average velocity of zero, since they end up back at the starting line.
Worked example: A cyclist rides 30 km in 1.5 hours along a winding trail, then returns along the same trail (60 km round trip in 3 hours total). Average speed for the full trip is 60 km ÷ 3 h = 20 km/h. Average velocity, though, is total displacement (0 km, since they returned to the start) divided by time, which comes out to 0 km/h. The Speed Calculator computes the distance-over-time figure, while the Velocity Calculator uses displacement, so plugging the same trip into both makes the difference concrete.
Step 2: Acceleration and Changing Motion
Acceleration measures how quickly velocity itself changes, speeding up, slowing down, or turning, calculated as the change in velocity divided by the time over which that change occurs (a = Δv/Δt). A positive acceleration in the direction of motion means speeding up; a negative one (deceleration) means slowing down.
Worked example: A car accelerates from 0 to 27 m/s (about 60 mph) in 6 seconds. Acceleration = (27 − 0) / 6 = 4.5 m/s². The car's velocity increases by 4.5 meters per second for every second that passes. The Acceleration Calculator computes this directly from initial velocity, final velocity, and time, and can solve for any one of those three values given the other two.
A subtlety worth flagging here, and covered again in the FAQs: an object can have zero velocity at an instant while still accelerating. A ball thrown straight up sits motionless at its peak for just a moment, yet gravity never stops pulling on it.
Step 3: Newton's Second Law and Force
Newton's second law ties acceleration to its cause: force equals mass times acceleration (F = ma). This single equation explains why heavier objects resist accelerating, or stopping, more than lighter ones under the same applied force, and it underlies everything from rocket thrust calculations to braking distance on a highway.
Worked example: A 1,200 kg car needs to decelerate from 25 m/s to a stop in 5 seconds. Required deceleration = 25/5 = 5 m/s², and the force needed is F = 1,200 kg × 5 m/s² = 6,000 N. That's the force the brakes, via friction with the road, must generate to stop the car in that time. The Force Calculator solves for force, mass, or acceleration given the other two values, covering both this braking scenario and the reverse case of finding acceleration from a known applied force.
Step 4: Momentum and Impulse
Momentum (p = mv) captures how much "quantity of motion" an object carries, folding mass and velocity into a single vector. It matters because it's always conserved in any collision or interaction between objects. The total momentum before a collision equals the total momentum after, even when kinetic energy gets lost to heat or deformation.
Impulse is the change in momentum, equal to force multiplied by the time over which that force acts (J = FΔt). This is why extending the duration of a collision, through airbags, crumple zones, or a padded landing, lowers the peak force experienced for the same overall change in momentum.
Worked example: A 0.15 kg baseball traveling at 40 m/s is caught and brought to a stop in 0.02 seconds. Its momentum is 0.15 × 40 = 6 kg·m/s, so the impulse needed to stop it is also 6 kg·m/s in the opposite direction. The average force on the catcher's glove is F = J/Δt = 6 / 0.02 = 300 N. If the catcher instead "gives" with the ball over 0.1 seconds, the force drops to 6/0.1 = 60 N, five times less, for the identical change in momentum. The Momentum Calculator and Impulse Calculator compute both quantities directly.
Step 5: Projectile Motion (Real-World Trajectories)
Projectile motion describes any object launched into the air and moving under gravity alone, ignoring air resistance, whether that's a thrown ball, a kicked football, or a fired artillery shell. Horizontal and vertical motion stay independent of each other: horizontal velocity remains constant throughout the flight, since no horizontal force acts on it, while vertical velocity changes continuously under gravity, producing the familiar parabolic arc.
Worked example: A ball is launched at 20 m/s at a 30-degree angle above the horizontal. Its horizontal velocity component is 20 × cos(30°) ≈ 17.3 m/s, constant throughout flight, and its vertical component is 20 × sin(30°) = 10 m/s, decelerating under gravity at 9.8 m/s². Time to reach peak height is 10/9.8 ≈ 1.02 seconds, total flight time is about 2.04 seconds, and horizontal range comes to roughly 17.3 × 2.04 ≈ 35.3 meters. The Projectile Motion Calculator computes range, maximum height, and flight time instantly from launch speed and angle, and lets you compare how changing the angle, holding speed constant, shifts all three results.
Step 6: Free Fall Under Gravity
Free fall is the special case of motion where gravity is the only force acting on an object, no air resistance, no other forces, governed by three formulas: velocity after falling for time t is v = gt, distance fallen is d = ½gt², and velocity after falling a distance d is v = √(2gd), where g is 9.8 m/s² on Earth.
Worked example: An object dropped from a 45-meter height, with no initial velocity, takes t = √(2×45/9.8) ≈ 3.03 seconds to hit the ground, reaching a final velocity of v = 9.8 × 3.03 ≈ 29.7 m/s (about 107 km/h) just before impact. The Free Fall Calculator computes fall time, final velocity, and distance fallen from any one known value. As the FAQs cover, this result doesn't depend on the object's mass at all, since gravitational acceleration is the same regardless of how heavy the object is.
Real falls of light or large-surface-area objects diverge from this idealized model once air resistance grows large enough to produce a terminal velocity, at which point the object stops accelerating altogether.
Putting It Together: Kinematics vs Dynamics
The first two steps here, speed/velocity and acceleration, belong to kinematics. They describe motion purely in terms of position, time, and rates of change, without asking what causes it. Force, momentum, and impulse belong to dynamics, the branch that explains why objects move the way they do by connecting motion to mass and the forces acting on it. Projectile motion and free fall combine both: kinematics describes the parabolic path, while gravity, a force governed by Newton's second law, shapes that path in the first place.
This layered structure is why textbooks and physics courses tend to teach these topics in the order used here. Speed and velocity come first since they're easiest to visualize, then acceleration as the natural next step, then force as the explanation for acceleration, then momentum and impulse for interactions between objects, and finally the two combined scenarios students and engineers actually calculate by hand most often: something thrown at an angle, and something falling straight down.
A useful sanity check when working any kinematics problem is unit consistency. Mixing km/h with meters and seconds is the single most common source of calculation errors. Convert everything to consistent SI units (meters, seconds, kilograms, newtons) before applying any formula, and you avoid the vast majority of mistakes. Every calculator linked in this guide handles that conversion internally, so you can enter values in whichever units are convenient and trust the underlying math.
Common Mistakes to Avoid
A few errors show up repeatedly when people first work through these formulas by hand. Confusing average velocity with average speed on a round trip is one of the most common, since a return to the starting point always yields zero net displacement even after covering real distance. Forgetting that vertical and horizontal motion are independent in projectile problems is another; students often try to apply a single formula to the whole trajectory rather than treating the horizontal (constant velocity) and vertical (constant acceleration) components separately. And assuming heavier objects fall faster remains one of the most persistent physics misconceptions, directly contradicted by the free-fall formulas above, which contain no mass term at all. Gravitational acceleration is the same for every object in the absence of air resistance.
Key Terms
- Momentum: the product of an object's mass and velocity, a vector quantity conserved in collisions
- Newton's Second Law: the principle that force equals mass times acceleration (F = ma), linking force to the motion it produces
- Velocity: the rate of change of displacement with respect to time, including direction, distinct from speed
- Acceleration: the rate of change of velocity with respect to time, whether speeding up, slowing down, or changing direction
- Impulse: the change in momentum an object experiences, equal to force multiplied by the time it acts
- Projectile Motion: the curved path of an object launched into the air and moving under gravity alone, with independent horizontal and vertical components
- Terminal Velocity: the constant speed a falling object reaches once air resistance balances the force of gravity