Time, Velocity, and Speed
Figure 1 The motion of these racing snails can be described by their speeds and their velocities. (credit: tobitasflickr, Flickr)
By the end of this section, you will be able to:
There is more to motion than distance and displacement. Questions such as, “How long does a foot race take?” and “What was the runner's speed?” cannot be answered without an understanding of other concepts. In this section, we add definitions of time, velocity, and speed to expand our description of motion.
As discussed in Physical Quantities and Units, the most fundamental physical quantities are defined by how they are measured. This is the case with time. Every measurement of time involves measuring a change in some physical quantity. It may be a number on a digital clock, a heartbeat, or the position of the Sun in the sky. In physics, the definition of time is simple—time is change, or the interval over which change occurs. It is impossible to know that time has passed unless something changes.
The amount of time or change is calibrated by comparison with a standard. The SI unit for time is the second, abbreviated s. We might, for example, observe that a certain pendulum makes one full swing every 0.75 s. We could then use the pendulum to measure time by counting its swings or, of course, by connecting the pendulum to a clock mechanism that registers time on a dial. This allows us to not only measure the amount of time, but also to determine a sequence of events.
How does time relate to motion? We are usually interested in elapsed time for a particular motion, such as how long it takes an airplane passenger to get from his seat to the back of the plane. To find elapsed time, we note the time at the beginning and end of the motion and subtract the two. For example, a lecture may start at 11:00 A.M. and end at 11:50 A.M., so that the elapsed time would be 50 min. Elapsed time \(\Delta t\) is the difference between the ending time and beginning time,
where \(\Delta t\) is the change in time or elapsed time, tf is the time at the end of the motion, and \(t_0\) is the time at the beginning of the motion. (As usual, the delta symbol, \(\Delta\), means the change in the quantity that follows it.)
Life is simpler if the beginning time \(t_0\) is taken to be zero, as when we use a stopwatch. If we were using a stopwatch, it would simply read zero at the start of the lecture and 50 min at the end. If \(t_0\)=0, then \(\Delta t=t_f\equiv t\).
In this text, for simplicity's sake,
Give an example (but not one from the text) of a device used to measure time and identify what change in that device indicates a change in time.
What does the symbol \(\Delta\) means?
Your notion of velocity is probably the same as its scientific definition. You know that if you have a large displacement in a small amount of time you have a large velocity, and that velocity has units of distance divided by time, such as miles per hour or kilometers per hour.
Average velocity is displacement (change in position) divided by the time of travel,
where \(\bar{v}\) is the average (indicated by the bar over the \(\bar{v}\)) velocity, \(\Delta x\) is the change in position (or displacement), and \(x_f\) and \(x_0\) are the final and beginning positions at times tf and \(t_0\), respectively. If the starting time \(t_0\) is taken to be zero, then the average velocity is simply
Notice that this definition indicates that velocity is a vector because displacement is a vector. It has both magnitude and direction. The SI unit for velocity is meters per second or m/s, but many other units, such as km/h, mi/h (also written as mph), and cm/s, are in common use. Suppose, for example, an airplane passenger took 5 seconds to move −4 m (the minus sign indicates that displacement is toward the back of the plane). His average velocity would be
The minus sign indicates the average velocity is also toward the rear of the plane.
The average velocity of an object does not tell us anything about what happens to it between the starting point and ending point, however. For example, we cannot tell from average velocity whether the airplane passenger stops momentarily or backs up before he goes to the back of the plane. To get more details, we must consider smaller segments of the trip over smaller time intervals.
Figure 1 A more detailed record of an airplane passenger heading toward the back of the plane, showing smaller segments of his trip.
The smaller the time intervals considered in a motion, the more detailed the information. When we carry this process to its logical conclusion, we are left with an infinitesimally small interval. Over such an interval, the average velocity becomes the instantaneous velocity or the velocity at a specific instant. A car's speedometer, for example, shows the magnitude (but not the direction) of the instantaneous velocity of the car. (Police give tickets based on instantaneous velocity, but when calculating how long it will take to get from one place to another on a road trip, you need to use average velocity.) Instantaneous velocity \(v\) is the average velocity at a specific instant in time (or over an infinitesimally small time interval).
Mathematically, finding instantaneous velocity,\(v\), at a precise instant \(t\) can involve taking a limit, a calculus operation beyond the scope of this text. However, under many circumstances, we can find precise values for instantaneous velocity without calculus.
Object G moves faster than object H. Which of the objects would cover more distance at the same amount of time?
A car travels 827 km in 4 hours. Compute the distance, in kilometers, traveled after 7 hours.
Two boys covered the same distance after running. Boy 1 recorded his time as 120 seconds and boy 2 has 90 seconds. Compare the velocity of the two boys.
Find the velocity, in \(m \over s\), of a particle that travels 330 meters in 2 hours.
Arrange the velocity of the cars from least to greatest if they are covering a distance of 5 km, 4 km and 10 km at the same time. Name the cars X, Y and Z respectively.
Two trains are approaching from opposite directions, separated by a distance of 7 km. The first train is traveling at 51 \(km \over h\) and the second one at 64 \(km \over h\). Calculate the time in which both trains will meet. Express your answer in seconds.
Alex, Man, and Pon are planning to have a race upon going home. If Alex can run 2 m in 10 s, Pon can run 5 m in 30 s, and Man can run 0.5 m every second, who possibly win?
A child playing with marbles makes a precise shot to win. If the child's hand is 96 cm from the target and his shot takes 0.2 seconds to reach it, how fast does the marble leave the child's hand?
Express your answer in meters per second.
Car A and car B both have the same velocity. If car A travels for 1 hour and car B travels for half more hour, which car traveled more distance?
50 km/h North, 10 m/s East and 2 mi/min South describe what quantity?
A car travels at 44 \(km \over h\) during 47 minutes to the north, turns to the east, and travels at 21 \(km \over h\) during 13 minutes, then finally turns to the south and goes at 68 \(km \over h\) during 21 minutes. Compute the magnitude of the average velocity and express your answer in kilometers per hour.
In everyday language, most people use the terms “speed” and “velocity” interchangeably. In physics, however, they do not have the same meaning and they are distinct concepts. One major difference is that speed has no direction. Thus speed is a scalar. Just as we need to distinguish between instantaneous velocity and average velocity, we also need to distinguish between instantaneous speed and average speed.
Instantaneous speed is the magnitude of instantaneous velocity. For example, suppose the airplane passenger at one instant had an instantaneous velocity of −3.0 m/s (the minus meaning toward the rear of the plane). At that same time his instantaneous speed was 3.0 m/s. Or suppose that at one time during a shopping trip your instantaneous velocity is 40 km/h due north. Your instantaneous speed at that instant would be 40 km/h—the same magnitude but without a direction. Average speed, however, is very different from average velocity. Average speed is the distance traveled divided by elapsed time.
We have noted that distance traveled can be greater than displacement. So average speed can be greater than average velocity, which is displacement divided by time. For example, if you drive to a store and return home in half an hour, and your car's odometer shows the total distance traveled was 6 km, then your average speed was 12 km/h. Your average velocity, however, was zero, because your displacement for the round trip is zero. (Displacement is change in position and, thus, is zero for a round trip.) Thus average speed is not simply the magnitude of average velocity.
Figure 1 During a 30-minute round trip to the store, the total distance traveled is 6 km. The average speed is 12 km/h. The displacement for the round trip is zero, since there was no net change in position. Thus the average velocity is zero.
Another way of visualizing the motion of an object is to use a graph. A plot of position or of velocity as a function of time can be very useful. For example, for this trip to the store, the position, velocity, and speed-vs.-time graphs are displayed in Figure 1. (Note that these graphs depict a very simplified model of the trip. We are assuming that speed is constant during the trip, which is unrealistic given that we'll probably stop at the store. But for simplicity's sake, we will model it with no stops or changes in speed. We are also assuming that the route between the store and the house is a perfectly straight line.)
Figure 2. Position vs. time, velocity vs. time, and speed vs. time on a trip. Note that the velocity for the return trip is negative.
If you have spent much time driving, you probably have a good sense of speeds between about 10 and 70 miles per hour. But what are these in meters per second? What do we mean when we say that something is moving at 10 m/s? To get a better sense of what these values really mean, do some observations and calculations on your own:
___________________is the magnitude of velocity.
The speed of propagation of the action potential (an electrical signal) in a nerve cell depends (inversely) on the diameter of the axon (nerve fiber). If the nerve cell connecting the spinal cord to your feet is 1.5 m long, and the nerve impulse speed is 18 \(m \over s\), how long does it take for the nerve signal to travel this distance?
Express your answer in seconds.
A 1 micrometer long bacteria moves the distance the equivalent of 46 times its size in 1 hour. Calculate the distance traveled by the bacteria. Express the result in micrometers.
Calculate the speed of the bacteria. Express the result in \({m \over s}\).
Every car has a speedometer that also has an odometer which records the distance travelled by the car. If the reading in the odometer after half an hour is 40 km, which gives the car’s average speed? Suppose the car started from rest.
Is it possible for the velocity of a particle to change if its speed is constant? Or changes its speed if the velocity is constant? Explain your reasoning. If yes, give an example.
The speed of propagation of the action potential (an electrical signal) in a nerve cell depends (inversely) on the diameter of the axon (nerve fiber). If the nerve cell connecting the spinal cord to your feet is 1.4 m long, and the nerve impulse speed is 18 \(m \over s\), how long does it take for the nerve signal to travel this distance? Express your answer in seconds.
Izrafel drove his bicycle from home, round to the park, then back home with a speed of 3.8 m/s. What is his average velocity?
A cheetah can travel 793 m in 26 seconds. What is its average speed in meters per second?
Average speed is not the magnitude of average velocity.
The speed of propagation of the action potential (an electrical signal) in a nerve cell depends (inversely) on the diameter of the axon (nerve fiber). If the nerve cell connecting the spinal cord to your feet is 1.5 m long, and the nerve impulse speed is 17 \(m \over s\), how long does it take for the nerve signal to travel this distance?
The gunman "Lazy" McAndrew assaults the state bank and runs away to the border on his mustang reaching a speed of 33 km/h. After 17 minutes, the sheriff arrived at the crime scene and undertakes the pursuit.
How many minutes will take "Lazy" McAndrew to arrive at the border if it is 24 km away from the bank?
Calculate the minimum speed in kilometers per hour that the sheriff needs to reach McAndrew before he crosses the border.
Richard walks 23 minutes at 4 \(m \over s\) every day. Compute the distance, in meters, traveled by Richard in a solar year.
A woman drives at 42 \(km \over h\) to a meeting out of her office. If she takes the same route back to the office and wants to maintain an average speed of 65 \(km \over h\) for the whole trip, at what speed, in kilometers per hour, should she be driving from the meeting to the office?
A delivery must arrive to its destination at 4:00 PM. If the delivery truck leaves the office at 11:00 AM, and its destination is 708 km away, at what speed, in \(mi \over h\), should the delivery truck be traveling to arrive on time?
A space agency is planning to send two spaceships to the International Space Station (ISS). The first spaceship travels at a speed of 5.1 \(km \over s\) and the second at 3.4 \(km \over s\), both traveling at a constant speed. If the first spaceship travels for 76 seconds, how many minutes later will the second spaceship arrive at the ISS?
Consider the speed of light through the air as \(3.00\times10^8 \; {m \over s}\) and the speed of sound through the air as \(300 \; {m \over s}\). How long does it take for an observer to hear the thunder if he is 138 km away after observing the light? Express your answer in seconds.
A lady goes to the store on her bicycle to buy some milk and travels 0.5 km in 5 minutes, then she walks 1.5 km in 12 minutes. What is the average speed, in \(km \over h\), of the lady when doing the route?
For her daily training, an athlete runs a distance of 299 m in 1 minute. What is her average speed in meters per second?
How many seconds does it take for a cat running with a speed of 1.1 \(m \over s\) to travel 6 meters?
A car traveling at 30 \(\frac ms\) decelerates at a constant rate to a complete stop after traveling 48 m. What is the average speed of the car?
How many seconds does it take for the car to stop?
This is a large modal.
