Kinematics in Two Dimensions; Vectors

Vectors and Scalars

In physics, quantities are classified as either vectors or scalars. Understanding the distinction is fundamental for analyzing motion and forces.

• Vector: A quantity that has both magnitude and direction.

• Examples of vectors: displacement, velocity, force, momentum.

• Scalar: A quantity that has only magnitude (no direction).

• Examples of scalars: mass, time, temperature.

Addition of Vectors—Graphical Methods

Vector addition is essential for combining physical quantities. The method used depends on the number of dimensions involved.

• One Dimension: Simple arithmetic addition and subtraction, paying attention to signs (direction).

• Two Dimensions: Vectors can be added using graphical methods such as:

  • Tail-to-tip method: Place the tail of the second vector at the tip of the first; the resultant vector is drawn from the tail of the first to the tip of the second.

  • Parallelogram method: Both vectors start at the same point; the resultant is the diagonal of the parallelogram formed.

• Pythagorean Theorem: If vectors are at right angles, the resultant magnitude is found using .

• Order of Addition: Vector addition is commutative; the order does not affect the result.

Subtraction of Vectors, and Multiplication of a Vector by a Scalar

Vectors can be subtracted and multiplied by scalars, which is important for analyzing changes and scaling physical quantities.

• Subtraction: To subtract vector B from A, add the negative of B (same magnitude, opposite direction): .

• Multiplication by a Scalar: Multiplying a vector V by a scalar c gives a vector with magnitude and direction:

  • If c is positive, direction is unchanged.

  • If c is negative, direction is reversed.

Adding Vectors by Components

Vectors can be broken down into perpendicular components, usually along the x and y axes, to simplify calculations.

• Component Form: Any vector can be written as , where and are the x and y components.

• Finding Components: Use trigonometric functions if the vector makes an angle with the x-axis:

• Adding Components: Add corresponding components arithmetically:

• Resultant Vector: Find magnitude and direction:

  • Magnitude:

  • Direction:

• Procedure:

  1. Draw a diagram; add vectors graphically.

  2. Choose x and y axes.

  3. Resolve each vector into x and y components.

  4. Calculate each component using sines and cosines.

  5. Add the components in each direction.

  6. Find the length and direction of the resultant vector.

Projectile Motion

Projectile motion describes the path of an object moving in two dimensions under the influence of gravity. The path is a parabola.

• Definition: A projectile is any object moving near Earth's surface under gravity, with negligible air resistance.

• Analysis: Separate the motion into horizontal (x) and vertical (y) components.

• Horizontal Motion: Constant velocity ( is constant).

• Vertical Motion: Constant acceleration ().

• Example: Two balls dropped simultaneously: one falls vertically, the other has an initial horizontal velocity. Both have identical vertical positions at identical times; the horizontal position of the moving ball increases linearly.

• Launch at an Angle: If launched at angle with initial speed :

  • Horizontal component:

  • Vertical component:

Solving Projectile Motion Problems

Projectile motion problems involve constant acceleration in two dimensions. The acceleration is downward.

• Procedure:

  1. Read the problem carefully; choose the object(s) to analyze.

  2. Draw a diagram.

  3. Choose an origin and coordinate system.

  4. Decide on the time interval; same in both directions, includes only the time with constant acceleration .

  5. Examine x and y motions separately.

  6. List known and unknown quantities.

  7. Remember: never changes; at the highest point, .

  8. Plan the solution; use appropriate equations, possibly combining them.

• Kinematic Equations for Projectile Motion:

  • Horizontal motion:

  • Vertical motion:

  • Vertical velocity:

  • At highest point:

• Table: Kinematic Equations for Projectile Motion

Additional info: If y is taken positive downward, the minus signs in front of g become plus signs.

Projectile Motion Is Parabolic

Projectile motion follows a parabolic path, which can be demonstrated mathematically by expressing y as a function of x.

• Equation: By eliminating time from the kinematic equations, the path is:

• This is the equation of a parabola.

Relative Velocity

Relative velocity describes how the velocity of an object appears from different reference frames. In two dimensions, velocities are added or subtracted as vectors.

• Notation: is the velocity of object A relative to object B.

• Example: For a boat moving in water:

  • : velocity of water relative to shore

  • : velocity of boat relative to shore

  • : velocity of boat relative to water

• Relationship:

• Velocities are added as vectors.

Summary of Chapter 3

• A quantity with magnitude and direction is a vector.

• A quantity with magnitude but no direction is a scalar.

• Vector addition can be done graphically or using components; the sum is called the resultant vector.

• Projectile motion is the motion of an object near Earth's surface under gravity.