Scalar and Vector Quantities Flashcards

Scalar & Vector Quantities

Scalar Quantities

• A scalar is any quantity that has a magnitude but no specified direction.

• A scalar quantity can be negative; the minus sign indicates a point on a scale.

• Examples of scalar quantities:

  • A person's height of 1.8 m

  • 1698 kJ of energy per 100 g of sugar

  • A speed limit of 120 km/h

  • An average winter temperature of -10ºC in Russia

Vector Quantities

• A vector is any quantity with both magnitude and direction.

• Examples of vector quantities:

  • A velocity of 90 km/h east

  • A force of 500 newtons north

• The direction of a vector in one-dimensional motion is given by a plus (+) or minus (−) sign.

• Graphically, vectors are drawn as arrows.

• An arrow has both a magnitude (length) and a direction (the direction in which the tip is pointing).

• Examples:

  • +3 m east

  • −2 m west

  • −1 m west

Vector Examples in 2-D

• A = 2.8 km, 45° North of the East axis

• B = 3.2 km, 22.5° East of the South axis

• C = 2.8 km, 45° North of the East axis

• The magnitude is calculated as $(\sqrt{(𝑥)^2+(𝑦)^2})$.

• Examples with coordinates:

  • (2;2)

  • (1;-3)

  • (2;2)

• Quadrants:

  • (+) (+)

  • (-) (-)

Vector Addition

• To add vectors graphically, they must be placed tip to tail.

• The resultant vector points from the tail of the 1st vector to the tip of the 2nd vector.

• If the direction toward the right is considered positive (+) and left as negative (-).

• Resultant vector is the sum of vectors, e.g., $100N = 400N + (-300N)$.

• Example: Walking 5m East, 8m North, then 3m West results in a position 8m North and 2m East of the starting point.

• The diagonal vector connecting the starting position with the final position is called the resultant vector.

• Tip-to-tail technique of vector addition.

Distance, Distance Travelled, & Displacement

• Distance is defined as the magnitude or size of displacement between two positions.

• Distance travelled is the total length of the path covered between two positions.

• Displacement is the change in position relative to a reference point.

• ∆𝑥 = 𝑥𝑓 − 𝑥𝑖

• $where$ ∆𝑥$is the displacement, and$𝑥𝑓$and$𝑥𝑖$are final and initial positions, respectively.</p></li><li><p>Position is where the object is at any particular time with respect to a reference point.</p></li></ul><h4 id="c6859b99-47ff-481f-9055-ccdc093ee786" data-toc-id="c6859b99-47ff-481f-9055-ccdc093ee786" collapsed="false" seolevelmigrated="true">Example 1</h4><ul><li><p>Scenario: Running from home to a shop that is 3km away and returning home in 45 minutes.</p></li><li><p>Reference point: Home</p></li><li><p>(a) Distance traveled and displacement while at the shop:</p><ul><li><p>Distance traveled: 3 km</p></li><li><p>Distance: 3 km</p></li><li><p>Displacement: 3 km East</p></li></ul></li><li><p>(b) Distance travelled and displacement after returning home:</p><ul><li><p>Distance travelled: 6 km (total distance covered)</p></li><li><p>Distance: 0 km</p></li><li><p>Displacement: 0 km (since the final and initial positions are the same)</p></li></ul></li><li><p>Note: Distance between two positions is not the same as the distance travelled between them.</p></li></ul><h4 id="244ea5bf-d82e-4c9a-ad07-cd0604a6dee8" data-toc-id="244ea5bf-d82e-4c9a-ad07-cd0604a6dee8" collapsed="false" seolevelmigrated="true">Example 2</h4><ul><li><p>An ant walks 2 meters West, 3 meters North, and 6 meters East.</p></li><li><p>Reference point: (0;0)</p></li><li><p>X1 = (-2;0) m, X2 = (0;3) m and X3 = (6;0) m</p></li><li><p>(a) The new position of an ant:</p><ul><li><p>$𝑿𝟏 + 𝑿𝟐 + 𝑿𝟑 = (-2 + 0 + 6, 0 + 3 + 0) m = (4; 3) m$</p></li><li><p>The new position is 4 meter East and 3 meter North.</p></li></ul></li><li><p>(b) The displacement of the ant:</p><ul><li><p>$\sqrt{(4)^2+(3)^2} = 5 m$$

• 5 m North of the East