Speed and Velocity - Ninth Grade Science
Having established how to quantify an object's spatial location using distance and displacement, the next logical step in mechanics is to investigate how rapidly that position changes over time. In everyday conversation, the words "speed" and "velocity" are treated as interchangeable synonyms. In physics, however, confusing them introduces profound errors into kinematic equations. Speed describes how fast a body covers distance regardless of direction, whereas velocity specifies both how fast the body travels and the exact direction in which it heads. This chapter explores how to measure, calculate, and graphically interpret uniform speed, non-uniform speed, instantaneous velocity, and average velocity across various physical scenarios.
Speed: The Rate of Distance Traversed
Speed is defined as the time rate at which an object covers distance along its path. Because distance is a scalar quantity containing no directional information, speed is likewise a scalar quantity. It informs you of the rapidity of motion but tells you nothing about where the object is heading.
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| Speed at a Glance |
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| Definition: Distance traveled per unit time |
| Formula: Speed = Total Distance / Time Elapsed (v = d / t) |
| Nature: Scalar quantity (Magnitude only, always >= 0) |
| SI Unit: Meters per second (m/s) |
| Common: Kilometers per hour (km/h), Miles per hour (mph) |
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Units of Speed and Unit Conversions
The standard SI unit of speed is meters per second (m/s). In everyday life and transportation, speed is commonly expressed in kilometers per hour (km/h). To convert from kilometers per hour to meters per second, multiply by five and divide by eighteen (since 1 km = 1000 m and 1 hour = 3600 s, $1000 / 3600 = 5 / 18$ or approximately 0.2778):
v \text{ (m/s)} = v \text{ (km/h)} \times \frac{5}{18}
Conversely, to convert from meters per second to kilometers per hour, multiply by eighteen and divide by five (or multiply by 3.6). For example, a highway speed limit of 90 km/h converts to:
90 \times \frac{5}{18} = 25\text{ m/s}
Instantaneous Speed versus Average Speed
In real-world journeys, moving bodies rarely maintain a constant speed. A school bus stops at red lights, speeds up along open boulevards, and slows down around tight turns. Instantaneous speed is the speed of an object at an infinitesimally small instant in time, precisely what an automobile's speedometer displays at any given second. In contrast, average speed is the total distance traveled divided by the total time elapsed for the entire journey:
v_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time Elapsed}} = \frac{d_{\text{total}}}{t_{\text{total}}}
Notice that average speed is not simply the mathematical mean of different speeds unless the object traveled at those speeds for identical durations of time.
Velocity: The Rate of Displacement
Velocity is defined as the time rate of change of displacement of an object. Because displacement is a vector quantity possessing both magnitude and direction, velocity is fundamentally a vector quantity. To state an object's velocity completely, you must specify both its speed (the magnitude of velocity) and its direction of motion.
Speed: 25 m/s (Scalar: Fast, but direction is unknown)
Velocity: 25 m/s Due North (Vector: Magnitude + Direction)
^ North
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| Velocity Vector = 25 m/s
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o (Moving Body)
Mathematical Formulation of Average Velocity
Average velocity ($\vec{v}_{ ext{avg}}$) is calculated as the net displacement divided by the total time interval:
\vec{v}_{\text{avg}} = \frac{\Delta \vec{x}}{\Delta t} = \frac{x_{\text{final}} - x_{\text{initial}}}{t_{\text{final}} - t_{\text{initial}}}
Because displacement can be positive, negative, or zero in one-dimensional kinematics, velocity can also be positive, negative, or zero. An object traveling in the negative coordinate direction has a negative velocity, indicating that its position coordinate is decreasing over time.
How Velocity Can Change
A common misconception is that an object's velocity changes only when its speed increases or decreases. Because velocity is a vector, it changes whenever its magnitude changes, its direction changes, or both change simultaneously. A car driving around a circular racetrack at a constant speed of sixty kilometers per hour is constantly accelerating because its direction of motion changes continuously at every point along the curve, meaning its velocity is constantly changing despite steady speed.
Average Speed versus Average Velocity: Critical Discrepancies
Because distance can exceed displacement, the average speed of a journey often differs dramatically from the magnitude of the average velocity.
Point A (Start) ------------------------> Point B (Turnaround)
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Returns to Point A
Total Distance = 100 m + 100 m = 200 m
Total Displacement = 0 m
Time Elapsed = 40 s
Average Speed = 200 m / 40 s = 5.0 m/s
Average Velocity = 0 m / 40 s = 0.0 m/s
The Closed-Circuit Journey
Consider an Olympic runner who completes a 400-meter oval track in 50 seconds, returning to the exact starting line. The runner's average speed is:
v_{\text{avg}} = \frac{400\text{ m}}{50\text{ s}} = 8.0\text{ m/s}
However, because the runner's starting and ending positions are identical, the net displacement is zero meters. Therefore, the runner's average velocity over the entire 50-second interval is:
\vec{v}_{\text{avg}} = \frac{0\text{ m}}{50\text{ s}} = 0.0\text{ m/s}
This contrast demonstrates that average velocity does not reflect the mechanical effort expended along the path; it reflects solely the net spatial displacement achieved per unit time.
Uniform and Non-Uniform Velocity
Kinematics categorizes motion based on whether velocity remains steady or varies across time.
Uniform Velocity
An object is in uniform velocity if it covers equal displacements in equal intervals of time, no matter how small those intervals may be. For an object to possess uniform velocity, two conditions must be satisfied simultaneously: it must travel at a perfectly constant speed, and it must move along an unwavering, perfectly straight line. Any deviation in speed or direction destroys uniformity.
Non-Uniform (Variable) Velocity
An object possesses non-uniform velocity if it covers unequal displacements in equal intervals of time, or if its direction of travel fluctuates. Virtually all everyday motion—from pedestrian traffic to aircraft navigating turbulence—is non-uniform. Analyzing non-uniform velocity requires differential calculus or graphical analysis to determine instantaneous rates of change.
Chapter Exercises
Exercise 1: An express train travels the first 120 kilometers of a journey at an average speed of 60 kilometers per hour, and the next 120 kilometers at an average speed of 80 kilometers per hour in the same direction. Calculate the total time required for the journey and determine the average speed of the train across the entire 240-kilometer trip. Explain why the average speed is not simply the arithmetic mean of 60 and 80.
Exercise 2: A swimmer swims across a 50-meter swimming pool from south to north in 25 seconds, turns around instantly, and swims back to the starting edge in 35 seconds. Calculate the swimmer's average speed for the round trip, and calculate the swimmer's average velocity for the entire swim.
Exercise 3: A passenger vehicle travels along a circular path of radius 70 meters at a constant speed of 11 meters per second. Determine the time required for the vehicle to complete one full revolution. State the vehicle's instantaneous speed, instantaneous velocity, average speed, and average velocity upon completing one full revolution.
Solutions and Explanations
Solution 1: To find the total time, calculate the time elapsed for each segment separately. For the first 120 km at 60 km/h, time $t_1 = 120 / 60 = 2.0$ hours. For the second 120 km at 80 km/h, time $t_2 = 120 / 80 = 1.5$ hours. The total distance is 120 plus 120, which equals 240 kilometers, and the total time is 2.0 plus 1.5, which equals 3.5 hours. The average speed is total distance divided by total time: 240 divided by 3.5, which equals approximately 68.57 kilometers per hour. The average speed is lower than the simple arithmetic mean of 70 km/h because the train spent more time traveling at the slower speed (2.0 hours) than at the faster speed (1.5 hours), giving greater weighting to the slower speed in the time-averaged calculation.
Solution 2: The total distance traveled by the swimmer is 50 meters forward plus 50 meters return, totaling 100 meters. The total time elapsed is 25 seconds plus 35 seconds, which equals 60 seconds. The average speed is total distance divided by total time: 100 meters divided by 60 seconds, which equals approximately 1.67 meters per second. For average velocity, we determine net displacement. Because the swimmer finishes at the exact starting edge, net displacement is 0 meters. Average velocity is net displacement divided by total time: 0 meters divided by 60 seconds, which equals 0.0 meters per second.
Solution 3: The circumference of the circular path is given by $2 imes \pi imes r = 2 imes (22/7) imes 70 = 440$ meters. At a constant speed of 11 meters per second, the time required for one full revolution is distance divided by speed: 440 divided by 11, which equals 40 seconds. At any given moment, the vehicle's instantaneous speed is 11 meters per second. Its instantaneous velocity has a magnitude of 11 meters per second directed tangent to the circular path at that specific location. For the entire single revolution, the average speed is total distance divided by time: 440 meters divided by 40 seconds, which equals 11 meters per second. The average velocity for one complete revolution is zero meters per second, because the vehicle returns to its exact starting coordinates, resulting in zero net displacement.