Physics Quiz Study Guide: Motion, Distance, Speed, and Velocity

Scalar vs. Vector Quantities

• Core Definitions:

  • Measurement types convey specific information based on whether they describe scalar or vector quantities.

  • Scalar Quantity: Describes amount (magnitude) only. A scalar tells how much, how fast, or how far an object moves, but does not specify a direction.

  • Examples of scalar quantities:

    • $10\,\text{meters}$

    • $5\,\text{seconds}$

    • $20\,\text{m/s}$

    • $3\,\text{kilometers}$

  • Vector Quantity: Describes both amount (magnitude) and direction. A vector tells how much and which way an object moves.

  • Examples of vector quantities:

    • $10\,\text{meters east}$

    • $5\,\text{m/s north}$

    • $20\,\text{meters to the left}$

    • $3\,\text{m/s downward}$

• Memory Trick:

  • Scalar = just a number and unit.

  • Vector = number, unit, and direction.

• Test Tip:

  • If a question provides a direction, such as north, south, left, or right, it is describing a vector quantity.

  • Be careful: some vector quantities can be negative or positive to show direction along a chosen axis, even when the word "direction" is not written out explicitly.

Distance vs. Displacement

• Distance ($\text{Total path traveled}$):

  • Measures the entire route an object travels, no matter which direction it goes.

  • It is a scalar quantity.

  • It is always zero or positive.

  • Measured in meters ($\text{m}$), kilometers ($\text{km}$), etc.

• Displacement ($\text{Change in position}$):

  • Measures how far and in which direction an object is from where it started.

  • It is a vector quantity.

  • It can be positive, negative, or zero, depending on the chosen direction.

  • Measured in meters ($\text{m}$), kilometers ($\text{km}$), etc.

• Example: Walking to School and Back

  • Imagine walking $6\,\text{meters east}$, then turning around and walking $2\,\text{meters west}$.

  • Distance Calculation:

  • $6 + 2 = 8\,\text{m}$

  • Total path = $8\,\text{meters}$

  • Displacement Calculation:

  • $6 - 2 = 4\,\text{m east}$

  • Final position = $4\,\text{meters east of start}$

• Calculations and Formulas:

  • Distance Formula:

  • $\text{Distance} = \text{Add all parts of the path}$

  • Example: Walk $3\,\text{m north}$, then $4\,\text{m south}$.

  • Calculation: $3 + 4 = 7\,\text{m}$

  • Displacement Formula:

  • $\text{Displacement} = \text{Final position} - \text{Starting position}$

  • Example: Walk $3\,\text{m north}$, then $4\,\text{m south}$. Take north as positive ($+$).

  • Calculation: $-1 - 0 = -1\,\text{m}$

  • Meaning: $1\,\text{meter south}$ of the starting point.

• Important Rule to Remember:

  • If you walk in a circle and return to where you started, your distance is not zero, but your displacement is zero.

Speed vs. Velocity

• Comparison:

  • Both concepts describe how quickly an object moves. The difference is whether direction matters.

  • Speed:

  • How fast something moves (simply how fast traveling).

  • Scalar quantity.

  • No direction needed.

  • Uses distance and time.

  • Example: A car travels at $60\,\text{km/h}$.

  • Velocity:

  • How fast something moves and in which direction (speed in a given direction).

  • Vector quantity.

  • Includes direction.

  • Uses displacement and time.

  • Example: A car travels at $60\,\text{km/h east}$.

• Formulas:

  • Average Speed:

  • $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

  • Tells how much distance is traveled per unit of time.

  • Average Velocity:

  • $\text{Velocity} = \frac{\text{Displacement}}{\text{Time}}$

  • Tells how much displacement occurs per unit of time.

Rearranging Formulas and Worked Examples

• Formula Triangle Method:

  • Use a triangle to remember the relationship between distance ($D$), speed ($S$), and time ($T$):

  • Distance: $D = S \times T$

  • Speed: $S = \frac{D}{T}$

  • Time: $T = \frac{D}{S}$

  • For velocity, use the exact same rearranging process, replacing distance with displacement and speed with velocity.

• Worked Examples:

  • Example A: Find Speed

  • Problem: A student runs $100\,\text{meters}$ in $20\,\text{seconds}$. How fast is the student running?

  • Step 1: Write the formula: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

  • Step 2: Put in the numbers: $\frac{100}{20} = 5$

  • Answer: $5\,\text{m/s}$

  • Example B: Find Distance

  • Problem: A car travels at $15\,\text{m/s}$ for $8\,\text{seconds}$. How far does it travel?

  • Step 1: Rearrange the formula: $\text{Distance} = \text{Speed} \times \text{Time}$

  • Step 2: Multiply: $15 \times 8 = 120$

  • Answer: $120\,\text{m}$

  • Example C: Find Time

  • Problem: A runner travels $60\,\text{meters}$ at a speed of $5\,\text{m/s}$. How long does it take?

  • Step 1: Rearrange the formula: $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

  • Step 2: Divide: $\frac{60}{5} = 12$

  • Answer: $12\,\text{seconds}$

  • Example D: Find Velocity

  • Problem: A person moves $30\,\text{meters east}$ in $10\,\text{seconds}$. What is the velocity?

  • Calculation: $\text{Velocity} = \frac{30\,\text{m east}}{10\,\text{s}}$

  • Answer: $3\,\text{m/s east}$

• Important Test Tips for Problem Solving:

  • Speed uses total distance.

  • Velocity uses displacement.

  • Always include units, such as $\text{m/s}$.

  • Always include direction for velocity.

Position vs. Time Graphs and Slope Calculations

• Understanding Graph Axes and Slope:

  • A position vs. time graph shows where an object is at different times.

  • The horizontal axis (x-axis) shows time.

  • The vertical axis (y-axis) shows position.

  • The slope tells you the object's velocity (how quickly position changes over time and in which direction).

• Slope Formula:

  • $\text{Slope} = \frac{\text{Change in position}}{\text{Change in time}}$

  • $v = \frac{x_2 - x_1}{t_2 - t_1}$

  • This is also the formula for average velocity between two points.

• Worked Example: Calculating the Slope

  • Problem: An object moves from position $2\,\text{meters}$ at $1\,\text{second}$ to position $10\,\text{meters}$ at $5\,\text{seconds}$.

  • Step 1: Find the change in position: $10 - 2 = 8\,\text{m}$

  • Step 2: Find the change in time: $5 - 1 = 4\,\text{s}$

  • Step 3: Divide: $\text{Slope} = \frac{8}{4} = 2\,\text{m/s}$

  • Answer: $2\,\text{m/s}$ (the object moves at a constant velocity of $2\,\text{m/s}$ in the positive direction).

• Slope Recognition Rules:

  • Flat line = At rest: Position is not changing. Slope $= 0$, Velocity = 0$.\n * **Upward slope = Positive velocity:** Position is increasing over time.\n * **Downward slope = Negative velocity:** Position is decreasing over time.\n * **Steeper line = Greater speed magnitude:** The larger the slope's absolute value, the faster the object moves.\n * **Straight line = Constant velocity:** A straight line can have a positive or negative slope; both represent constant velocity.\n\n# Motion Categorization on Position vs. Time Graphs\n\n* **Object At Rest:**\n * An object at rest stays in the same position as time passes.\n * Example: A student sits in a chair for 5\,\text{seconds} without moving.\n * Key observations:\n * The line is flat.\n * The position stays at 5\,\text{meters}.\n * The slope is zero.\n * Velocity is zero.\n\n\n\n* **Object Moving at Constant Positive Velocity:**\n * Changes position by the same amount during each equal time interval.\n * Example: A person walks forward 2\,\text{meters} every second.\n * Key observations:\n * The line is straight and slanted upward.\n * Position increases by 2\,\text{meters} every second.\n * The slope is 2\,\text{m/s}.\n * Velocity is constant and positive.\n\n\n\n* **Object Moving at Constant Negative Velocity:**\n * Moves in the negative direction at a constant velocity.\n * Example: Position decreases from 10\,\text{meters}$to$0\,\text{meters}$over$5\,\text{seconds}.\n * Key observations:\n * The line is straight and slopes downward.\n * Position decreases by 2\,\text{meters} every second.\n * The slope is -2\,\text{m/s}.\n * Velocity is constant and negative.\n\n\n\n* **Critical Graph Warnings ("Do Not Confuse the Graphs!"):**\n * A horizontal line does not mean the object is at position zero. It means the object's position is not changing.\n * A downward line does not mean the object is slowing down. A straight downward line means it is moving at a constant velocity in the negative direction.\n * A position vs. time graph is not a picture of the spatial path the object takes. It shows position as time passes.\n\n# Essential Core Summaries and Test Strategies\n\n* **Concept Review Table:**\n * Scalar: Amount only\n * Vector: Amount and direction\n * Distance: Total path traveled\n * Displacement: Final position -$ starting position

  • Speed: Distance $\div$ time

  • Velocity: Displacement $\div$ time

  • Slope: Change in position $\div$ change in time

  • Flat position-time line: Object at rest

  • Straight upward line: Constant positive velocity

  • Straight downward line: Constant negative velocity

  • Steeper slope: Greater velocity magnitude

• The 10-Second Memory Trick:

  • Distance = the whole trip.

  • Displacement = where you ended up compared with where you started.

  • Speed = how fast.

  • Velocity = how fast and which way.

  • Slope = velocity.

  • Flat graph = stopped.

  • Straight slanted graph = constant velocity.

• Test-Taking Strategy (3 Questions to Ask Before Solving):

  1. Am I measuring the whole path or the change in position?

  2. Does direction matter?

  3. Am I being asked to calculate a number or interpret a graph?

  • Answering these three questions first helps select the correct formula and prevents confusing distance, displacement, speed, and velocity.

Practice Questions & Detailed Explanations

• Question 1: A car travels at $40\,\text{m/s}$. Is this scalar or vector?

  • Options: Scalar | Vector

  • Answer: Scalar

  • Explanation: Speed tells how fast something moves but does not give a direction.

• Question 2: A student walks $5\,\text{m north}$. Is this distance or displacement?

  • Options: Distance | Displacement | Both

  • Answer: Both

  • Explanation: The student travels $5\,\text{meters}$ (distance) and ends up $5\,\text{meters north}$ of the starting point (displacement).

• Question 3: A runner travels $80\,\text{meters}$ in $10\,\text{seconds}$. What is the average speed?

  • Options: $8\,\text{m/s}$ | $800\,\text{m/s}$ | $0.125\,\text{m/s}$

  • Answer: $8\,\text{m/s}$

  • Explanation: $80 \div 10 = 8\,\text{m/s}$

• Question 4: A student walks $10\,\text{m east}$ and then $4\,\text{m west}$. What is the distance?

  • Options: $6\,\text{m}$ | $14\,\text{m}$ | $10\,\text{m}$

  • Answer: $14\,\text{m}$

  • Explanation: $10 + 4 = 14\,\text{m}$. Distance is the total path traveled.

• Question 5: Using the same trip ($10\,\text{m east}$ then $4\,\text{m west}$), what is the displacement?

  • Options: $14\,\text{m east}$ | $6\,\text{m east}$ | $6\,\text{m west}$

  • Answer: $6\,\text{m east}$

  • Explanation: $10 - 4 = 6\,\text{m east}$. Displacement is the final position compared with the starting position.

• Question 6: An object moves $24\,\text{m east}$ in $6\,\text{seconds}$. What is its velocity?

  • Options: $4\,\text{m/s}$ | $4\,\text{m/s east}$ | $144\,\text{m/s east}$

  • Answer: $4\,\text{m/s east}$

  • Explanation: $24 \div 6 = 4\,\text{m/s east}$. Velocity includes direction.

• Question 7: What does the slope of a position vs. time graph represent?

  • Options: Distance | Speed only | Velocity

  • Answer: Velocity

  • Explanation: Slope measures the change in position divided by the change in time.

• Question 8: What does a flat horizontal line on a position vs. time graph mean?

  • Options: Constant positive velocity | Object at rest | Constant negative velocity

  • Answer: Object at rest

  • Explanation: The object's position stays the same as time passes. Its velocity is zero.

• Question 9: What does a straight line sloping upward on a position vs. time graph represent?

  • Options: Constant positive velocity | Object at rest | Constant negative velocity

  • Answer: Constant positive velocity

  • Explanation: The object's position increases by the same amount during each equal time interval.

• Question 10: An object changes position from $4\,\text{m}$ to $16\,\text{m}$ in $3\,\text{seconds}$. What is the slope?

  • Options: $4\,\text{m/s}$ | $12\,\text{m/s}$ | $20\,\text{m/s}$

  • Answer: $4\,\text{m/s}$

  • Explanation: $\text{Slope} = \frac{16 - 4}{3} = \frac{12}{3} = 4\,\text{m/s}$