Understanding Vectors, Motion, and Newton’s Laws

Posted on Apr 5, 2025 in Physics

Scalar Product

The scalar product of two vectors, a and b, results in a scalar value, k. This is obtained by multiplying the magnitudes of the two vectors and the cosine of the angle between them. The formula is expressed as:

k = |a| |b| cos(α)

For the vectors to be perpendicular, their scalar product must be zero. The formula can be rearranged to solve for the cosine of the angle:

cos(α) = (a · b) / (|a| |b|)

Vector Product

The vector product of two vectors, a and b, results in a new vector with the following attributes:

  • Magnitude: |a x b| = |a| |b| sin(α)
  • Direction: Perpendicular to the plane containing a and b.
  • Sense: Determined by the right-hand rule.

Vector Position, Velocity, and Instantaneous Acceleration

Vector Position

A body’s position can be communicated in two ways:

  • Cartesian Coordinates: (x, y)
  • Polar Coordinates: (r, θ)

The position vector, r, defines the location of a body with respect to a reference point and is measured in meters (m).

Vector Velocity

Velocity is defined as the rate of change of the position vector of a body. The velocity vector has the same direction and sense as the displacement. Instantaneous velocity is the limit of the average velocity as the time interval approaches zero, obtained by differentiating the position equation. The instantaneous velocity vector is tangent to the trajectory at any given point and indicates the direction of motion, measured in m/s.

Vector Acceleration

Acceleration measures how quickly velocity changes, measured in m/s2. Instantaneous acceleration is the limit of the change in velocity over an infinitesimally small time interval.

Intrinsic Components of Acceleration

Tangential Acceleration

Tangential acceleration produces changes in the magnitude of velocity. It can be defined as follows:

  • Magnitude: The rate of change of the magnitude of velocity.
  • Direction: Tangent to the trajectory at the point of interest.
  • Sense: In the direction of motion if the speed increases, and opposite to the motion if the speed decreases.

Centripetal Acceleration

Centripetal acceleration occurs in curvilinear motion and leads to changes in the direction of velocity without affecting its magnitude. It has the following characteristics:

  • Magnitude: ac = v2 / r, where v is the velocity and r is the radius of curvature.
  • Direction: Always directed towards the center of the curve.

Uniform Rectilinear Motion (URM)

The equation representing uniform rectilinear motion can be generalized as:

x = x0 ± vt

A positive sign indicates the body is moving away, while a negative sign indicates the opposite.

Uniformly Accelerated Rectilinear Motion (MRUA)

The equation that informs us of the values of velocity as a function of time for rectilinear motion with constant acceleration is:

v = v0 + at

The general equation that informs us of the position versus time for a body moving with constant acceleration is:

x = x0 + v0t + (1/2)at2

Newton’s Laws of Motion

Law of Inertia (Newton’s First Law)

An isolated body remains at rest or continues to move at a constant velocity unless acted upon by an external force. Reference systems at rest or moving at constant velocity are called inertial reference systems, where the principle of inertia and the laws of physics hold true.

Law of Force and Interaction (Newton’s Second Law)

Any interaction between two bodies results in a force. The effect of any interaction is the modification of velocity and, therefore, acceleration.

Law of Action and Reaction (Newton’s Third Law)

When two bodies interact, they exert equal and opposite forces on each other. These forces of action and reaction act on different bodies; otherwise, it would be impossible to change the motion of bodies.