Speed and Velocity

SPEED

• Speed is how fast an object changes its location.
• Speed is always a distance divided by a time: v=rac{d}{t}.
• Common units: miles per hour, meters per second, kilometers per hour, inches per minute, etc.
• SI unit is meters per second: $ext{m s}^{-1}$.
• Average speed is the total distance traveled divided by the total time: ext{average speed}=rac{ ext{total distance}}{ ext{total time}}.
• Rate is a general concept defined as one quantity divided by another quantity; examples:
  • gallons per minute
  • pesos per dollar
  • points per game
• Relationship: average speed is the rate at which distance is covered over time.

INSTANTANEOUS SPEED

• Instantaneous speed is the speed at a precise instant in time.
• It is the rate at which distance is being covered at a given instant.
• It is found by calculating the average speed over a short enough time interval during which the speed does not change much: $v_{ ext{inst}}<br /> oughly\approx \frac{\Delta d}{\Delta t}$ for very small \Delta t.
• Conceptual note: a speedometer provides instantaneous speed (no direction).

INSTANTANEOUS SPEED ILLUSTRATION

• A speedometer typically has scales (e.g., mph and km/h) to show instantaneous speed on a vehicle.
• This reflects that instantaneous speed is a local measurement, not a property of motion over an extended interval.

VELOCITY

• Definition: velocity is the rate of change of displacement: oldsymbol{v}=rac{\Delta \boldsymbol{r}}{\Delta t}. (displacement is the straight‑line change in position, including direction)
• Velocity is a vector quantity; it has both magnitude and direction.
• Displacement (\Delta \boldsymbol{r}) is different from distance traveled; displacement is the net change in position.
• Examples:
  • If displacement is 12 m in 4 s toward the east, then velocity magnitude is |oldsymbol{v}|=\frac{12\text{ m}}{4\text{ s}}=3\text{ m s}^{-1} toward the east.
• Direction is required to specify velocity; speedometer does not provide direction.

VELOCITY PROPERTIES

• Velocity includes both magnitude (speed) and direction.
• A change in velocity can be a change in speed or a change in direction (or both).
• Speedometer measures instantaneous speed only, not velocity.

CHANGING VELOCITY

• A change in velocity requires a force (Newtonian intuition): a force is needed to change either the magnitude or the direction of velocity.
• Examples:
  • For a car rounding a curve, friction between wheels and road provides the force to change direction.
  • For a ball bouncing off a wall, the wall exerts a force on the ball, changing its direction.

EXAMPLE: ARUNNER DISTANCE COMPOSITION

• Problem: A runner runs 200 m east, then 300 m west. The entire trip takes 50 s. Find the average speed and the average velocity.
• Step 1: Total distance (for speed):
  • Distance = 200 m + 300 m = 500 m.
  • Average speed: ext{avg speed}=rac{500\text{ m}}{50\text{ s}}=10\text{ m s}^{-1}.
  • Note: direction does not affect average speed.
• Step 2: Net displacement (for velocity):
  • Initial position: xi = 0; after moves, final position: xf = -100 m (east 200 m then 300 m west).
  • Net displacement: $\Delta x = x<em>f - x</em>i = -100\text{ m}.$
  • Average velocity (vector): $\boldsymbol{v}_{avg} = \frac{\Delta x}{\Delta t} = \frac{-100\text{ m}}{50\text{ s}} = -2\text{ m s}^{-1} \text{ (west)}.$
• Summary: Average speed = 10 m s^{-1}; Average velocity is 2 m s^{-1} toward the west (negative x direction).

AVERAGE SPEED VS AVERAGE VELOCITY (EXAMPLE 1 CONTINUED)

• Average speed depends on total distance covered and total time: ext{avg speed} = \frac{500\text{ m}}{50\text{ s}} = 10\text{ m s}^{-1}.$n- Average velocity depends on net displacement over total time: \boldsymbol{v}_{avg} = \frac{\Delta \boldsymbol{x}}{\Delta t} = \frac{-100\text{ m}}{50\text{ s}} = -2\text{ m s}^{-1} \text{ (west)}.$</li> </ul> <h3 id="keyconceptsandnotationsummary">KEY CONCEPTS AND NOTATION SUMMARY</h3> <ul> <li>Speed: magnitude only; scalar quantity.$v=\frac{d}{t}.$</li> <li>Velocity: displacement over time; vector quantity.$\boldsymbol{v}=\frac{\Delta \boldsymbol{r}}{\Delta t}.$</li> <li>Instantaneous speed: approximately$v_{inst} \approx \frac{\Delta d}{\Delta t}$for small time intervals.</li> <li>Average speed:$\text{avg speed}=\frac{\text{total distance}}{\text{total time}}.$</li> <li>Average velocity:$\boldsymbol{v}_{avg}=\frac{\Delta \boldsymbol{r}}{\Delta t}$over the full interval; includes direction.</li> <li>Directionality matters for velocity but not for speed.</li> </ul> <h3 id="practicalrealworldrelevance">PRACTICAL REAL-WORLD RELEVANCE</h3> <ul> <li>Distinguishing speed and velocity is essential for navigation, sports performance, and physics analyses.</li> <li>Direction is critical when planning motion (e.g., steering a vehicle, predicting trajectory).</li> </ul> <h3 id="formulastoremember">FORMULAS TO REMEMBER</h3> <ul> <li>Speed:$v=\frac{d}{t}$</li> <li>Instantaneous speed (approximate):$v_{inst} \approx \frac{\Delta d}{\Delta t}$</li> <li>Velocity (displacement-based):$\boldsymbol{v}=\frac{\Delta \boldsymbol{r}}{\Delta t}$</li> <li>Average speed:$\overline{v} = \frac{\text{total distance}}{\text{total time}}$</li> <li>Average velocity:$\boldsymbol{v}_{avg}=\frac{\Delta \boldsymbol{r}}{\Delta t}$$

CONNECTIONS TO FOUNDATIONAL PRINCIPLES

• Rate concepts (distance/time, displacement/time) underpin kinematics.
• Vector vs scalar quantities: magnitude vs magnitude+direction; forces produce changes in velocity via acceleration.

PRACTICAL CHECKLIST

• When asked for velocity, report both magnitude and direction.
• When asked for speed, report only the magnitude unless direction is specified.
• Use net displacement rather than total distance for velocity calculations.