Notes on Scalars, Vectors, and Motion Concepts

Scalars and Vectors

• A physical quantity described by a single number with a unit is a scalar quantity.
• A vector quantity has both a magnitude and a direction.
• The magnitude of a vector is its length.
• A vector is graphically represented as an arrow.

Displacement Vectors

• The displacement vector represents the distance and direction of an object’s motion.
• It is drawn from the object's initial position to its final position, regardless of the actual path taken between these points.

Tactics Box 1.3: Subtracting Vectors

• The net displacement for a trip with two legs is the sum of the two displacements that make it up.
• To add vectors:
  1. Draw the first vector.
  2. Place the tail of the second vector at the tip of the first.
  3. Draw an arrow from the tail of the first vector to the tip of the second.
• This process yields the vector sum (net displacement).
• Reference: PhET: Vectors and Motion (Text: page 20).

Vectors and Trigonometry

• Trigonometry is used to calculate lengths and angles of triangles and is applied to displacement vectors and other vectors.
• With displacement vectors, right triangles help find magnitude and direction using trigonometric functions.

Example 1.7: How Far Away Is Anna? (1 of 6)

• Scenario: Anna walks 90 m due east and then 50 m due north.
• Question: What is her displacement from the starting point?
• Strategy: Use a right triangle; set up a coordinate system with the origin at Anna’s start; represent the two motions as displacement vectors connected tail-to-head.
• Setup: Displacements are treated as vectors that can be added (vector sum).

Example 1.7: How Far Away Is Anna? (2 of 6) – Solve

• The two displacement vectors are placed head-to-tail to form a resultant vector, which represents Anna’s net displacement from the origin.
• The vector from the origin to the final point is the net displacement.

Example 1.7: How Far Away Is Anna? (3 of 6) – Solve

• Anna’s distance from the origin equals the length (magnitude) of the resultant vector, which is the hypotenuse of a right triangle with legs 50 m (north) and 90 m (east).
• So the hypotenuse is the net displacement magnitude.

Example 1.7: How Far Away Is Anna? (4 of 6) – Solve

• Magnitude of the net displacement is found with the Pythagorean theorem:
  $| ext{net displacement}| = \sqrt{(90\text{ m})^2 + (50\text{ m})^2}.$

Example 1.7: How Far Away Is Anna? (5 of 6) – Solve

• Rounding to appropriate significant figures gives $| ext{net displacement}| \approx 100\ \text{m}.$
• Direction: The angle north of east is given by
  $\theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right) = \tan^{-1}\left(\frac{50}{90}\right) \approx 29^{\circ}.$
• Therefore, net displacement ≈ 100 m at ~29° north of east.

Example 1.7: How Far Away Is Anna? (6 of 6) – Assess

• With legs of 50 m and 90 m, a hypotenuse of about 100 m seems reasonable.
• The angle is less than 45°, around 29°.

Velocity Vectors

• A velocity vector points in the direction of the object's motion; its magnitude is the object's speed.
• The velocity vector can be drawn on a motion diagram by connecting successive positions with vectors.

Example 1.8: Drawing a Ball’s Motion Diagram (1 of 4)

• Problem: Jake hits a ball at a 60° angle from the horizontal. It is caught by Jim.
• Task: Draw a motion diagram showing velocity vectors (not displacement vectors).
• Strategy: Problems often require reasonable interpretation of where motion begins and ends; here, assume the ball leaves the bat and ends just as it touches Jim’s hand.

Example 1.8: Drawing a Ball’s Motion Diagram (2 of 4) – Prepare

• Model the ball as a particle and sketch a motion diagram representing its path along an arc.
• The diagram should begin with the ball in motion.

Example 1.8: Drawing a Ball’s Motion Diagram (3 of 4) – Solve

• The ball is hit at 60° to the horizontal; velocity vectors are shown by connecting successive dots with arrows.
• Observations: Vectors become shorter (ball slowing down) and then longer (ball speeding up) and the direction changes.
• Conclusion: This is not constant-velocity motion.

Example 1.8: Drawing a Ball’s Motion Diagram (4 of 4) – Assess

• The diagram should reflect the ball moving upward at the start and downward at the end, matching a typical toss-back-and-forth motion.
• The quick assessment supports the reasonableness of the diagram.

Integrated Example 1.9: A Goose Gets Its Bearings (1 of 7)

• Concept: Migrating geese use landmarks, rivers, roads, and sun position to determine direction.
• Path: The goose flies in a straight line for part of the journey, then makes a corrective right-angle turn.

Integrated Example 1.9: A Goose Gets Its Bearings (2 of 7)

• After one hour, the goose rests at a lake that lies due east of its original position.
• Question: How much extra distance did the goose travel due to its initial erroneous direction?
• Focus: Compare the actual path length to the straight-line distance to the final position.

Integrated Example 1.9: A Goose Gets Its Bearings (3 of 7)

• Questions to address: What was the goose’s flight speed? A typical speed for a migrating goose is $80\ \text{km/h}.$ Does the result seem reasonable?
• Strategy: Combine right-triangle geometry with speed and significant figures.

Integrated Example 1.9: A Goose Gets Its Bearings (4 of 7)

• PREPARE: Redraw the figure to highlight the displacement from start to end; show the displacement as the hypotenuse of a right triangle.

Integrated Example 1.9: A Goose Gets Its Bearings (5 of 7)

• SOLVE: The minimum distance the goose could have flown straight to the lake is the hypotenuse of a triangle with legs $21\ \text{mi}$ and $28\ \text{mi}$, giving a straight-line distance of
  $\text{straight-line distance} = \sqrt{21^2 + 28^2} = 35\ \text{mi}.$
• The actual distance flown is the sum of the two legs: $21\ \text{mi} + 28\ \text{mi} = 49\ \text{mi}.$
• Extra distance flown: $49 - 35 = 14\ \text{mi}.$

Integrated Example 1.9: A Goose Gets Its Bearings (6 of 7)

• To compute flight speed, use
  $\text{speed} = \frac{\text{total distance flown}}{\text{total time}}.$
• To compare with typical goose speed, convert the solution to $\text{km/h}$: use 1 mile ≈ 1.61 km.

Integrated Example 1.9: A Goose Gets Its Bearings (7 of 7) – Assess

• The calculated flight speed should be close to the typical goose speed, supporting the result.
• The net displacement of $35\ \text{mi}$ should be reasonable as the hypotenuse of the right-triangle drawing.

Summary: Important Concepts (1 of 5) Motion Diagrams

• The particle model represents a moving object as if all its mass were at a single point.
• A motion diagram shows the object’s positions at successive times using dots.
• In a motion diagram, the time interval between successive dots is constant.
• Reference: Text: page 25.

Summary: Important Concepts (2 of 5) Scalars and Vectors

• Scalar quantities have only a magnitude (e.g., temperature, time, mass).
• Vector quantities have both magnitude and direction (e.g., velocity, displacement).
• Velocity vectors can be drawn on a motion diagram by connecting successive points with a vector.
• Reference: Text: page 22.

Summary: Important Concepts (3 of 5) Describing Motion

• Position locates an object relative to a chosen coordinate system and is described by a coordinate.
• A change in position is a displacement.
• For motion along a line, displacement is a signed quantity. The displacement from $x<em>i$ to $x</em>f$ is $\Delta x = x<em>f - x</em>i.$

Summary: Important Concepts (4 of 5) Describing Motion

• Time is measured from a particular instant to which we assign $t = 0$.
• A time interval is the elapsed time between two instants $t<em>i$ and $t</em>f$ and is given by $\Delta t = t<em>f - t</em>i.$
• Velocity is the ratio of the displacement to the time interval during which the displacement occurs: $v = \frac{\Delta x}{\Delta t}.$

Summary: Important Concepts (5 of 5) Units

• Every measurement must include a unit.
• The SI system is used in science.
• Common SI units:
  • Length: $\text{m}$
  • Time: $\text{s}$
  • Mass: $\text{kg}$

Summary: Applications (1 of 2) Working with Numbers

• Scientific notation expresses a number as a decimal between 1 and 10 multiplied by a power of ten.
• Example: the diameter of the Earth can be written in scientific notation; حيث earth diameter is often given in scientific notation as a decimal times a power of ten.
• Prefixes indicate multiples of 10 before a unit (e.g., $1\ \text{km} = 1000\ \text{m}$).
• Unit conversions use conversion factors, e.g., to convert km to miles:
  $1 \text{ mi} = 1.61\ \text{km} \, (approx).$

Summary: Applications (2 of 2) Working with Numbers

• Significant figures rules:
  • For multiplication, division, and powers, the result has the same number of significant figures as the factor with the fewest significant figures.
  • For addition and subtraction, the result is limited by the value with the smallest number of decimal places.
• An order-of-magnitude estimate is an estimate with roughly one significant figure, usually using rough, everyday values.