Unit Conversions

Introduction to Unit Conversions

Unit conversions are essential in physics to ensure that all quantities are expressed in compatible units for calculations. Conversion factors are used to translate measurements from one unit system to another.

• Definition: A unit conversion is the process of changing a quantity expressed in one set of units to another set using conversion factors.

• Example: Converting to :

• Application: Always check that units cancel appropriately and the final answer is in the desired units.

Vectors

Components and Projections of Vectors

Vectors are quantities that have both magnitude and direction. In physics, vectors are often described using their components along the Cartesian axes.

• Position Vector:

• General Vector:

• Magnitude:

• Unit Vector:

Scalar (Dot) Product

The dot product of two vectors yields a scalar and is useful for finding the angle between vectors.

• Formula:

• Angle Calculation:

• Application: To find the angle between two vectors, use the dot product and solve for .

Projection of a Vector

The projection of a vector onto an axis is the magnitude of the vector times the cosine of the angle between the vector and the axis. The sign depends on the direction relative to the axis.

• If the angle is measured with respect to the negative direction, a negative sign is included.

• In two dimensions, if the angle is measured counter-clockwise from the x-axis, the cosine projects on the x-axis and the sine on the y-axis. If measured clockwise from the y-axis, the cosine projects on the y-axis and the sine on the x-axis.

• If the sign of the y component is zero, direction is undefined.

Angle of a Vector with Respect to an Axis

For a two-dimensional vector specified by its Cartesian components, the angle with respect to the x-axis is:

Kinematics

Introduction to Kinematics

Kinematics is the study of motion without considering the forces that cause it. It involves analyzing displacement, velocity, and acceleration.

• Displacement:

• Velocity:

• Acceleration:

Motion with Constant Acceleration

When acceleration is constant, the following kinematic equations apply:

These equations can be applied separately to each component (x, y, z) in multi-dimensional motion.

Impact Conditions

Impact conditions describe the behavior of objects during collisions with surfaces or boundaries.

• Application: These conditions are used to solve for the time or position at which an object impacts a surface or reaches an extremum.

Additional info:

• In projectile motion, the acceleration in the x-direction is zero (), and in the y-direction is equal to gravity ().