Basics of Linear Motion Study Guide

Learning Intentions and Success Criteria

• WALT (We Are Learning To):

  • Identify vectors and scalars.

  • Describe distance and displacement.

  • Describe speed and velocity.

  • Describe acceleration.

• WILF (What I’m Looking For):

  • List 3 vectors and 3 scalars.

  • Calculate the distance and displacement travelled by an object.

  • State the difference between instant and average velocity.

  • Calculate acceleration.

Scalar vs. Vector Quantities

• Scalar Definition: A scalar is defined as a quantity that has a magnitude (size/value) but no direction.

• Vector Definition: A vector is defined as a quantity that has both magnitude and direction.

• Illustrative Example:

  • If a ball moves a distance of $2\,m$, this is a scalar value because it only describes the magnitude.

  • If a ball moves $2\,m$ to the left, this is displacement and is a vector because it specifies a direction.

Examples of Scalar Quantities

Scalar quantities are measured using numbers and units without directional context:

• Length: e.g., $16\,cm$

• Temperature: e.g., $102^{\circ}C$

• Time: e.g., $7\,s$

Examples of Vector Quantities

Vector quantities include a number, a unit, and a specific direction:

• Acceleration: e.g., $30\,m/s^{2}$ upwards.

• Displacement: e.g., $200\,miles$ northwest.

• Force: e.g., $2\,N$ downwards.

Vector Representation and Sign Conventions

• Vector Arrows: Arrows are used in physics to indicate both the direction and the sign (positive or negative) of a quantity.

• Coordinate Conventions:

  • Vertical: Upward directions are positive ($+$, North/N); downward directions are negative ($minus$, South/S).

  • Horizontal: Rightward or forward directions are positive ($+$, East/E); leftward or backward directions are negative ($minus$, West/W).

  • Compass Directions:

  • North ($N$): $+$

  • South ($S$): $-$

  • East ($E$): $+$

  • West ($W$): $-$

Adding Vectors in One Dimension Using Algebra

To determine the resultant vector, apply sign conventions to magnitudes based on their direction and then sum them algebraically.

• Example Case Study: A student walks $25\,m$ west, $16\,m$ east, $44\,m$ west, and then $12\,m$ east.

  • Step 1: Sign Assignment

  • $25\,m$ west = $-25\,m$

  • $16\,m$ east = $+16\,m$

  • $44\,m$ west = $-44\,m$

  • $12\,m$ east = $+12\,m$

  • Step 2: Calculation

  • Resultant vector = $(-25) + (+16) + (-44) + (+12) = -41\,m$

  • Step 3: Conclusion

  • Since the result is negative, the direction is west. Resultant vector = $41\,m$ west.

• Practice Problem (Try Yourself 6.1.2): Determine the resultant vector for a box with forces: $16\,N$ up, $22\,N$ down, $4\,N$ up, and $17\,N$ down.

Vector Subtraction and the Concept of Change

• Delta Symbol (\Delta): In physics, the Greek symbol $\Delta$ is used to describe a ’change’ in a value.

• Definition of Change: describes the result of the final state subtract the initial state.

• Change in Velocity Formula: $\Delta v = v_{final} - u_{initial}$

  • $v$: final velocity.

  • $u$: initial velocity.

Distance vs. Displacement

• Distance ($d$): A scalar quantity describing how far an object travels during its entire journey.

• Displacement ($s$): A vector quantity describing the change in position.

• Displacement Formula: $s = \text{final position} - \text{initial position}$

• Units: Both are measured in meters ($m$).

Calculation Example: The City Building Lift

A lift carries a passenger from the ground floor to the basement, then up to the top floor. The basement is $10\,m$ below the ground floor, and the top floor is $50\,m$ above the ground floor.

• a) Displacement from ground floor to basement: $-10\,m$ (or $10\,m$ down).

• b) Displacement from basement to top floor: Final position is $+50\,m$, initial was $-10\,m$. $\Delta s = 50 - (-10) = +60\,m$ (or $60\,m$ up).

• c) Total distance travelled during the entire trip: Down $10\,m$ then up $60\,m$. $d = 10 + 60 = 70\,m$.

• d) Total displacement for the entire trip: Final position is $+50\,m$ (top floor), initial position was $0\,m$ (ground floor). $s = 50 - 0 = +50\,m$ (up).

Speed and Velocity

• Speed: Defined as $\frac{\text{distance}}{\text{time}}$. It is a scalar quantity.

• Velocity: Defined as $\frac{\text{displacement}}{\text{time}}$. It is a vector quantity.

• Units: Both are measured in meters per second ($m/s$ or $m\,s^{-1}$).

Average vs. Instantaneous Measures

• Average Speed/Velocity: Gives an indication of how fast an object is moving over a specific time interval.

• Instantaneous Speed/Velocity: Measures how fast an object is traveling at a specific, precise point in time.

Unit Conversion: m/s to km/h

• To convert from $m/s$ to $km/h$: Multiply by $3.6$ ($\times 3.6$).

• To convert from $km/h$ to $m/s$: Divide by $3.6$ ($\div 3.6$).

World Record Speed Data:

• Racehorse: $19.7\,m\,s^{-1}$

• Luge: $43\,m\,s^{-1}$

• Cricket delivery: $44.7\,m\,s^{-1}$

• Waterskiing (barefoot): $68.3\,m\,s^{-1}$

• Tennis serve: $73.1\,m\,s^{-1}$

• Train: $161\,m\,s^{-1}$

Average Velocity and Speed Case Study: Sally the Athlete

Sally jogs $100.0\,m$ west in $20.0\,s$, then turns and walks $160.0\,m$ east in a further $45\,s$ before stopping.

• Data Analysis:

  • Displacement 1: $-100.0\,m$

  • Displacement 2: $+160.0\,m$

  • Total Time: $20.0\,s + 45\,s = 65\,s$

• Calculations:

  • a) Average Velocity ($m/s$): Total Displacement / Total Time = $((-100) + 160) / 65 = 60 / 65 \approx 0.92\,m/s$ east.

  • b) Magnitude of Average Velocity ($km/h$): $0.92 \times 3.6 \approx 3.32\,km/h$.

  • c) Average Speed ($m/s$): Total Distance / Total Time = $(100 + 160) / 65 = 260 / 65 = 4.0\,m/s$.

  • d) Average Speed ($km/h$): $4.0 \times 3.6 = 14.4\,km/h$.

Acceleration

• General Concept: If displacement describes change in position and velocity describes change in position with time, then acceleration describes the change in velocity with time.

• Classification: Vector quantity.

• Measuring Units: Meters per second per second ($m/s^{2}$ or $m\,s^{-2}$).

• Formula: $a = \frac{v - u}{t}$

  • $v$: final velocity.

  • $u$: initial velocity.

  • $t$: time.

Calculation Example: Golf Ball Rebound

A golf ball is dropped onto a concrete floor and strikes the floor at $9.0\,m\,s^{-1}$. It then rebounds at $7.0\,m\,s^{-1}$. The contact time with the floor is $35\,ms$.

• Identify Values (using sign convention: up is positive, down is negative):

  • $u = -9.0\,m/s$ (striking the floor downwards).

  • $v = +7.0\,m/s$ (rebounds upwards).

  • $t = 35\,ms = 0.035\,s$.

• Step-by-Step Calculation:

  • Change in velocity ($\Delta v$) = $v - u = 7.0 - (-9.0) = 16.0\,m/s$.

  • Average Acceleration ($a$) = $\frac{16.0\,m/s}{0.035\,s} \approx 457.14\,m/s^{2}$ (upwards).