Newtonian Mechanics Kinematics Velocity Acceleration Force Work Kinetic Potential Energy Momentum

# Newtonian Mechanics from First Principles

## 1. A map of motion

Mechanics is the part of physics that describes how objects move and why their motion changes. A rolling ball, a jumping person, a falling raindrop, and a satellite all obey the same basic ideas. The first step is to choose a reference frame: an agreed origin, a direction called positive, and a clock.

Position tells where an object is. In one dimension we can call the position d or x and measure it in metres. Position is not the same as distance travelled. If you walk 5 m east and then 5 m west, your travelled distance is 10 m but your displacement is zero because you finish where you began.

Displacement is final position minus initial position:

The triangle symbol Δ means “change in.” A negative displacement simply points in the chosen negative direction.

## 2. Speed and velocity

Average speed is total distance divided by elapsed time. Average velocity uses displacement, so it includes direction:

Velocity is measured in metres per second, written m/s. A car moving east at 12 m/s and a car moving west at 12 m/s have equal speeds but opposite velocities. On a position-versus-time graph, velocity is the slope. A steep rising line means a large positive velocity; a horizontal line means rest.

Instantaneous velocity is the velocity at one moment. A speedometer approximates its magnitude. Mathematically it is the limiting slope of the position graph:

For a beginner, it is enough to imagine using shorter and shorter time intervals around the moment of interest.

## 3. Acceleration

Acceleration measures how quickly velocity changes:

Its unit is m/s². Acceleration can change speed, direction, or both. A car turning at constant speed is accelerating because its velocity direction changes. Negative acceleration does not automatically mean slowing down. An object moving in the negative direction with negative acceleration speeds up. Always compare the signs of velocity and acceleration.

On a velocity-versus-time graph, acceleration is the slope. The area beneath that graph gives displacement. On an acceleration-versus-time graph, signed area gives the change in velocity.

## 4. Constant-acceleration equations

When acceleration is constant, velocity changes by the same amount each second. Starting from initial velocity v₀:

The average velocity during the interval is one half of the sum of the initial and final velocities. Multiplying by time gives displacement:

Eliminating time produces a third useful relation:

Here d means displacement from the starting point, not necessarily total path length. These equations apply only during an interval with constant acceleration. Before choosing one, list what is known, decide which direction is positive, and identify the unknown.

### Worked example: a bicycle

A bicycle starts at 3 m/s and accelerates at 2 m/s² for 4 s. Its final velocity is 11 m/s. Its displacement is 28 m because the original motion contributes 12 m and the acceleration contributes 16 m. A quick unit check confirms that acceleration times time has velocity units and acceleration times time squared has distance units.

## 5. Free fall and projectiles

Near Earth, objects in free fall have an approximately constant downward acceleration with magnitude g = 9.81 m/s². Ignoring air resistance, mass does not change this acceleration. A heavy ball and a light ball released together fall together.

For a projectile, horizontal and vertical motion can be studied separately. With no air resistance, horizontal acceleration is zero while vertical acceleration is downward:

The components of an initial launch speed v₀ at angle θ are v₀ cos θ horizontally and v₀ sin θ vertically. Gravity changes only the vertical component. This independence explains the curved parabolic path.

## 6. Force: the cause of acceleration

A force is a push or pull. Force is a vector, so arrows are useful: arrow direction shows force direction and arrow length represents magnitude. The SI unit is the newton. One newton accelerates one kilogram by one metre per second each second.

Common forces include weight, support or normal force, tension in a rope, friction along a surface, spring force, drag, and thrust. Weight near Earth is:

Mass measures inertia and is expressed in kilograms. Weight is a gravitational force and is expressed in newtons. Mass stays essentially constant when a person travels to the Moon; weight changes because the local gravitational field changes.

## 7. Newton's first law

An object remains at rest or moves with constant velocity unless a nonzero net external force acts on it. This is the law of inertia. Motion does not require a continuing net force; changing motion does. A puck gliding on nearly frictionless ice continues because very little horizontal force opposes it.

Equilibrium means that all forces add to zero. An object in equilibrium can be at rest or can travel at constant velocity:

## 8. Newton's second law

The vector sum of all external forces equals mass times acceleration:

This equation is a procedure. Choose the object, draw every external force acting on it, split forces into coordinate components, add the components, and solve. If a 2 kg cart experiences 10 N right and 4 N left, the net force is 6 N right and its acceleration is 3 m/s² right.

For two dimensions, apply the law separately:

## 9. Newton's third law

Whenever object A exerts a force on object B, object B exerts an equal-magnitude, opposite-direction force on object A. These two forces act on different objects, so they do not cancel on one free-body diagram. A swimmer pushes water backward; the water pushes the swimmer forward. A book pulls Earth upward gravitationally with the same magnitude that Earth pulls the book downward, although Earth's enormous mass makes its acceleration tiny.

## 10. Free-body diagrams

A free-body diagram isolates one object as a dot or box and shows only forces exerted on that object. Use this checklist:

1. Name the system being analysed.
2. Draw weight downward.
3. Add contact forces from every touching object.
4. Add tension, thrust, drag, or other real interactions.
5. Choose axes that simplify the geometry.
6. Never draw velocity as a force.

For a block on a ramp at angle θ, weight can be resolved into a component down the ramp and a component perpendicular to it:

If the block has no perpendicular acceleration, the normal force balances the perpendicular weight component.

## 11. Friction and springs

Static friction prevents slipping and adjusts up to a maximum. Kinetic friction acts after sliding begins:

Friction points opposite the relative sliding or the tendency to slide. It is not always opposite an object's velocity; a moving conveyor belt can use friction to speed a box forward.

An ideal spring follows Hooke's law within its elastic range:

The minus sign says the spring force points toward equilibrium. A larger spring constant k means a stiffer spring.

## 12. Work

Work transfers energy when a force acts through a displacement. For a constant force and straight displacement:

The angle θ is between force and displacement. A force parallel to motion does positive work, an opposing force does negative work, and a perpendicular force does zero work. Carrying a level backpack at constant height involves biological effort, but the upward support force does no mechanical work on the backpack because it is perpendicular to horizontal displacement.

Work is measured in joules. One joule equals one newton metre.

## 13. Kinetic energy and the work-energy theorem

The energy of motion is kinetic energy:

Because velocity is squared, doubling speed quadruples kinetic energy. The net work on an object equals its change in kinetic energy:

This theorem follows from Newton's second law and the constant-acceleration relations. It is often easier than solving for time.

## 14. Potential energy

Potential energy belongs to an interacting system. Near Earth's surface, gravitational potential energy relative to a chosen zero height is:

An ideal spring stores elastic potential energy:

Only changes in potential energy affect predictions, so the zero level may be chosen conveniently. Lifting an object increases gravitational potential energy; compressing or stretching a spring increases elastic potential energy.

## 15. Conservation of mechanical energy

If only conservative forces such as gravity and ideal spring forces do work, total mechanical energy remains constant:

For a dropped object, gravitational potential energy decreases while kinetic energy increases. When friction acts, mechanical energy is transformed into thermal energy. Total energy is still conserved, but the simple mechanical-energy total is not.

Power is the rate of energy transfer:

Power is measured in watts, or joules per second.

## 16. Momentum and impulse

Momentum combines mass and velocity:

Impulse is force applied over a time interval and equals the change in momentum:

Increasing stopping time reduces average force for the same momentum change. Airbags, helmets, and padded mats use this principle. On a force-versus-time graph, impulse is the signed area under the curve.

## 17. Conservation of momentum

For a system with negligible net external impulse, total momentum before an interaction equals total momentum afterward:

Internal interaction forces occur in Newton's third-law pairs and cancel in the system total. Momentum conservation works in every direction independently.

In an elastic collision, both momentum and kinetic energy are conserved. In an inelastic collision, momentum is conserved but some kinetic energy becomes sound, heat, or deformation. In a perfectly inelastic collision, objects stick together:

## 18. Circular motion and gravity

An object moving in a circle has acceleration toward the centre even at constant speed:

“Centripetal force” is not a new kind of force. It is the name for the inward net force supplied by tension, gravity, friction, or another interaction. Universal gravitation between masses M and m is:

An orbit is continuous free fall: gravity bends the object's velocity enough that it keeps missing the surface.

## 19. A reliable problem-solving method

First sketch the situation and choose a system. Write known values with units. Choose positive axes and keep the sign convention. Decide whether the problem is best described by kinematics, forces, energy, or momentum. Write a symbolic equation before inserting numbers. Solve algebraically, substitute units, and test whether the sign and scale make physical sense.

Dimensional analysis catches many mistakes. Terms added together must have the same units. An answer for energy should reduce to kg·m²/s², and an answer for force should reduce to kg·m/s².

## 20. Python particle-motion laboratory

The following program uses a small time step to simulate gravity, a spring, optional drag, and a floor. It plots a two-dimensional path and checks mechanical energy. Change the parameters and compare the numerical trajectory with constant-acceleration predictions. The semi-implicit update is more stable for spring motion than updating position first.

import numpy as np
import matplotlib.pyplot as plt

class Particle:
    def __init__(self, mass, position, velocity):
        self.m = float(mass)
        self.r = np.array(position, dtype=float)
        self.v = np.array(velocity, dtype=float)

def forces(p, gravity, spring_k=0.0, anchor=(0.0, 0.0), drag=0.0):
    weight = p.m * np.array(gravity, dtype=float)
    stretch = p.r - np.array(anchor, dtype=float)
    spring = -spring_k * stretch
    resistance = -drag * p.v
    return weight + spring + resistance

def simulate(p, duration=8.0, dt=0.002, gravity=(0.0, -9.81),
             spring_k=0.0, anchor=(0.0, 0.0), drag=0.0,
             floor=None, restitution=0.85):
    steps = int(duration / dt) + 1
    t = np.linspace(0.0, duration, steps)
    position = np.empty((steps, 2))
    velocity = np.empty((steps, 2))
    energy = np.empty(steps)
    anchor = np.array(anchor, dtype=float)

    for i, time in enumerate(t):
        position[i] = p.r
        velocity[i] = p.v
        kinetic = 0.5 * p.m * np.dot(p.v, p.v)
        gravitational = p.m * 9.81 * p.r[1]
        spring = 0.5 * spring_k * np.dot(p.r - anchor, p.r - anchor)
        energy[i] = kinetic + gravitational + spring

        acceleration = forces(p, gravity, spring_k, anchor, drag) / p.m
        p.v += acceleration * dt
        p.r += p.v * dt

        if floor is not None and p.r[1] < floor:
            p.r[1] = floor
            if p.v[1] < 0.0:
                p.v[1] *= -restitution

    return t, position, velocity, energy

particle = Particle(mass=1.0, position=(0.0, 2.0), velocity=(5.5, 8.0))
t, r, v, energy = simulate(particle, duration=3.0, floor=0.0, drag=0.03)

fig, axes = plt.subplots(1, 3, figsize=(14, 4))
axes[0].plot(r[:, 0], r[:, 1])
axes[0].set(xlabel="x (m)", ylabel="y (m)", title="Particle path")
axes[0].axis("equal")
axes[0].grid(True)

axes[1].plot(t, v[:, 0], label="vx")
axes[1].plot(t, v[:, 1], label="vy")
axes[1].set(xlabel="time (s)", ylabel="velocity (m/s)", title="Velocity")
axes[1].legend()
axes[1].grid(True)

axes[2].plot(t, energy)
axes[2].set(xlabel="time (s)", ylabel="mechanical energy (J)", title="Energy check")
axes[2].grid(True)
plt.tight_layout()
plt.show()

For an ideal no-drag flight without floor contact, the energy curve should be nearly flat. Reducing the time step should reduce numerical drift. With drag enabled, mechanical energy falls because energy is transferred to the surroundings.

## 21. Experiments to try

Measure a toy car's position at equal time intervals and estimate velocity from successive displacements. Film a tossed ball and compare its vertical positions with a quadratic curve. Hang masses from a spring, graph force against extension, and estimate the spring constant from slope. Roll carts together to compare total momentum before and after collision. Every experiment should include uncertainty: reaction time, ruler precision, friction, and air resistance explain differences between an ideal model and real data.

## 22. Common misconceptions

• A moving object does not need a net forward force to keep moving at constant velocity.
• Forces in a third-law pair act on different objects.
• Acceleration is not always in the direction of velocity.
• Normal force is not automatically equal to weight.
• Momentum can be conserved while kinetic energy is not.
• Energy is not “used up”; it changes form or moves between system and surroundings.

## 23. Final perspective

Mechanics becomes manageable when each symbol is tied to a physical story. Kinematics describes motion. Forces explain acceleration. Energy tracks transformations without following every instant. Momentum captures interactions over time. These are not separate worlds; they are complementary views of the same event. A strong solution states the system, assumptions, coordinates, governing principle, units, and reasonableness check.