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 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}}
oughly\approx \frac{\Delta d}{\Delta t} for very small \Delta t.
Conceptual note: a speedometer provides instantaneous speed (no direction).

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.

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 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.

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.

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 = xf - xi = -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 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)}.

Speed: magnitude only; scalar quantity. v=\frac{d}{t}.
Velocity: displacement over time; vector quantity. \boldsymbol{v}=\frac{\Delta \boldsymbol{r}}{\Delta t}.
Instantaneous speed: approximately v_{inst} \approx \frac{\Delta d}{\Delta t} for small time intervals.
Average speed: \text{avg speed}=\frac{\text{total distance}}{\text{total time}}.
Average velocity: \boldsymbol{v}_{avg}=\frac{\Delta \boldsymbol{r}}{\Delta t} over the full interval; includes direction.
Directionality matters for velocity but not for speed.

Distinguishing speed and velocity is essential for navigation, sports performance, and physics analyses.
Direction is critical when planning motion (e.g., steering a vehicle, predicting trajectory).

Speed: v=\frac{d}{t}
Instantaneous speed (approximate): v_{inst} \approx \frac{\Delta d}{\Delta t}
Velocity (displacement-based): \boldsymbol{v}=\frac{\Delta \boldsymbol{r}}{\Delta t}
Average speed: \overline{v} = \frac{\text{total distance}}{\text{total time}}
Average velocity: \boldsymbol{v}_{avg}=\frac{\Delta \boldsymbol{r}}{\Delta t}$$

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