Describing Motion Motion involves a change in the position of a body in space coupled with a change in time. Motion can be described in terms of speed, velocity, and acceleration, each of which has a distinct meaning in physics. Speed Speed is a measure of how "fast" a body is moving. Specifically, it is defined as the distance traveled by a body divided by the time required for the body to travel that distance. Symbolically, v = d/t where v means speed, d means distance, and t means time. (The letter v is used to represent speed because, as we shall soon see, speed is closely related to velocity). The equation above can be written in two alternative forms by solving for distance and for time: d = vt or t = d/v Speed can be measured in any combination of units of distance and time. Speedometers in automobiles, for example, are calibrated in units of both miles per hour (mi/h) and kilometers per hour (km/h). When the distance covered is short, speed measurements may be given in units of feet per second (ft/s) or meters per second. Meters per second (m/s) is the S1 unit for speed. It is a derived S1 unit, since it incorporates the S1 base units for distance and time.

Typical speeds of many moving things

Motion Approximate Speed (m/s) Approximate Speed (km/s) Light (in vacuum) 300,000,000 300,000

Sun around our galaxy 250,000 250 Earth around the sun 29,600 29.6 Moon around Earth 1,000 1.00

Commercial jet airliner 300 .30 Automobile (at 67 mi/h) 30 .03 Runner sprinting (max) 10 .01

Person walking 1.5 Snail crawling 0.01

Sperm swimming 0.00005

Since speed can change with time, it is important to distinguish between instantaneous speed and average speed. Instantaneous speed (the speed at any given instant of time) is distance divided by time for an infinitesimally small time interval. Average speed is distance divided by time for some arbitrary and usually not-so-small interval of time. The following example illustrates the difference between the two. Late at night, a person can drive from Toronto to Kitchener a distance of 110 mi, at a constant 55 mi/h. By applying Equation 5.lc, we find that the time of the trip is t = d/v

= 110 mi/(55 mi/h) =2h This result corresponds to the trip made by car A in the distance-versus time graph in If the same trip is made during the day, the driver will almost certainly get stuck in slow or stopped traffic along the way. Still, it may be possible (though not legal) to make it in 2 h. Car B, whose journey is also graphed in Figure 5.1, did this by speeding in certain areas where traffic was light. Owing to severe traffic congestion near Toronto at an utter standstill (v = 0) for about 10 min. Earlier, its maximum speed occurred between points m and n, 40 and 60 mi from Toronto. The straight line between those points on the graph indicates that car B's speed was constant through that stretch. Its speed there was v= d/t

= (60 mi - 40 mi)/(0.8 h - 0.6 h) = 20 mi/0.2 h = 100 mi/h

which is well above the posted 55-mi/h speed limit on the road between the two cities. For both trips, the average speed was 55 mi/h over the 2-h interval. The late-night driver's instantaneous speed always equaled the average speed; the daytime driver's instantaneous speeds were at different times less than, equal to, and greater than the average speed.

Velocity A body's speed within a reference frame only partially describes its motion. A more complete description includes the body's direction of motion as well. When we state both a body's speed and direction of motion, we are specifying its velocity. If we say that a person moves at 1 m/ s (a slow walking pace), we specify the person's speed only. If we say that a person walks 1 m/s south (or west, or southwest), we specify the person's velocity. Two objects with the same speed do not necessarily have the same velocity. A car moving at 60 mi/h north on Interstate 35 differs in velocity from another car moving at 60 mi/h south on the same road, though both cars share the same speed relative to the ground. Two baseballs, one moving up at 10 m/ s and the other moving down at 10 m/ s, differ in velocity as well. Notice in the examples just given how the stated direction of motion (north and south, up and down) is related to an implied reference frame. Just as speed must be related to a known reference frame, so too must the direction of a body's motion be related to a known reference direction or directions. It is often crucial to distinguish between instantaneous velocity and constant velocity. Instantaneous velocity is distance divided by time for an infinitesimally small time interval, plus information about the direction of motion during that same infinitesimally small time interval. If a body's instantaneous velocity does not change over time, then its velocity is constant. Constant velocity means constant speed and constant direction during a given (and not infinitesimally small) time interval. A body having a constant velocity is said to be in uniform motion Is it possible for a body to have a constant speed and yet not be moving uniformly? Try walking or running around an oval racetrack at a constant speed and ask yourself that question Your velocity is changing if you slow down or speed up while going in a straight line, if you change your direction while keeping the same speed, or if you change your speed and your direction at the same time. Vectors and Scalars Velocity is an example of a vector quantity. Speed is an example of a scalar quantity. A vector quantity, or vector, expresses both magnitude ("how much") and direction. A scalar quantity, or scalar, expresses magnitude only. Length (without any indication of direction), area, volume, mass, density, time, and temperature are all scalars because no direction is associated with them. Direction, in the context of our discussion, means direction in space. It may be tempting to think that temperature is a vector rather than a scalar because it goes "up" and "down." But temperature goes up and down only in a metaphorical sense. Temperature increases and decreases do cause mercury columns in standard thermometers to rise and fall, but

temperature itself heads nowhere. Time is another example of a scalar quantity with purely metaphorical directions (past and future) ascribed to it. Vectors are often represented by bold face type: for example, v, which means velocity (a vector), can be distinguished from v, which means speed (a scalar). We will soon make our acquaintance with another vector, force, which is denoted in type by the bold letter F. When force is symbolized by F, we are referring only to the magnitude of the force and not to its direction. On diagrams, vectors often appear as arrows whose lengths are proportional to the magnitudes of the quantities being represented. Differences between scalars and vectors become apparent when we add two or more of each. If we combine (add) two or more masses, they always add normally: 2 kg of water plus 3 kg of water always make a total of 5 kg of water. Vector quantities, on the other hand, add in a way that takes their direction into account. For example, the top of the treadmill belt, the velocity of a runner does not add to the velocity of the belt. The two velocities do not add; instead, the velocity is zero relative to the observer on the ground. Acceleration Velocity expresses all we need to know about a body's motion during an infinitesimally small interval of time. However, we can go one step further and describe how a body's motion changes with time, which is what physicists call acceleration. Acceleration is a change of velocity. You may have the idea that acceleration means only an increase in speed. You should abandon this notion! A body is accelerating if its speed is changing (increasing or decreasing), if its direction of motion is changing, or if both its speed and direction of motion are changing. (Remember that velocity specifies both speed and direction. If either or both of these things are changing, then there is acceleration.) Like velocity, acceleration is a vector quantity. To get a feeling for what acceleration means, try the following exercise, which is best performed outdoors or in a gymnasium: From rest, start walking in a straight line so that after 1 s your speed is 1 m/s (a slow walk). Gradually and smoothly, increase your pace so that after 2 s you are moving at 2 m/s (a brisk walk). At 3 s your speed should be 3 m/s (jogging pace). At 4 s, if you can still keep up, you should be moving at 4 m/s (a fast run). These increases in speed with time are graphed in Figure 5.5. A world-class sprinter would, by continuing in this manner, reach 10 m/s (about 22 mi/h) after 10 s. For as long as you engage in this exercise, your speed continually changes. But something else does remain constant: the ratio between the change in your speed and the time during which the change took place. This ratio is the magnitude of your acceleration (a), which is given by a = v/.t where v (read "delta v") is the change in speed over some time interval t (read "delta t"). Strictly speaking, this equation refers to the average acceleration over the time interval in question - though for this 'exercise it does not matter, since the acceleration is

assumed to be constant. There has been a v of 1 m/s for every1t of 1 s. Substituting into Equation 5.2, we get a = v / t = (1 m/s) / 1s = 1 m/s2 The unit m/s2 can be read as "meter(s) per second squared," but its meaning is "meter(s) per second per second." Note that we need only two of the four fundamental properties-distance and time (not mass or charge) - to express the magnitude of acceleration. Acceleration is the ratio between speed and time, and speed itself is derived from distance and time. In other words, we express acceleration by means of distance used once and time used twice. Now, let us consider the direction of acceleration. Three cases are shown in the diagram below. In every case, the magnitude of the acceleration is the same, 1 m/s2. In case (a), the acceleration and velocity vectors point in the same direction. In case (b), the runner is slowing down. His acceleration is in a direction opposite to the direction in which he is moving. The magnitude of his acceleration is still 1 m/s2 because his speed is changing by 1 m/s with every second of time. In case (c), a woman is moving around a small circular track. Her speed remains constant, but her direction of motion is steadily changing.

Acceleration involves a change in speed only, as in cases (a) and (b); a change in direction only, as in case (c); or a change in both speed and direction In case (c), the direction of the runner's velocity is changing uniformly because she is moving at a constant speed around a circle. Her acceleration vector is changing in such a way that it always points inward toward the center of the circular path The runners in (a) and (b) are undergoing linear acceleration, and the runner in (c) is undergoing centripetal acceleration.

Case (c) requires more explanation. Not only is the runner's direction of motion changing all the time, but also the direction of her acceleration is changing. As long as she maintains the same speed and does not deviate from the circular path, her acceleration vector keeps turning so that it always points inward toward the center of that path. Bodies moving on circular arcs at constant speed are said to be undergoing centripetal ("centerseeking") acceleration.The magnitude, a, of the centripetal acceleration is given by a = v2 / r where r is the radius of the circular path and v is the speed. The equation says that a and r are inversely related: Increasing the radius decreases the acceleration magnitude, and decreasing the radius increases the acceleration magnitude. For this reason, it is more difficult to make a sharp turn (a turn with a smaller radius of curvature) than a gradual turn. Note also that the v term in Equation 5.3 is squared. This means that a runner or a car moving through the same curve at twice the speed experiences four times the acceleration (22 = 4). In the case of the circular motion, values ofv = 3 m/s and r = 9 m produce a centripetal acceleration having a magnitude of a = v2 / r = (3 m/s)2 / 9 m = (9 m2 / s2) / 9 m = 1 m / s2 Other combinations of speed and radius could have yielded the same 1 m/ S2 acceleration: v = 1 m/s and r = 1 m, v = 2 m/s and r = 4 m, v = 16 m/s and r = 4 m, and many more. You can readily feel accelerations (or changes of motion) as long as their magnitudes are not too small. Step on the gas pedal of your car and you have the sensation of the seat pushing you forward. Hit the brakes hard and your tightly fastened seat belt "pushes" you back toward the seat. Turn the steering wheel right and you feel a push to the right. Turn the steering wheel left and you are forced to the left. When your car glides along a smooth, flat road in a straight line at constant speed, you feel very little sensation of movement, because there is very little change in your velocity (the magnitude of your acceleration is practically zero). On a road filled with potholes, every jerk up and down or back and forth lets you know that your velocity is changing in a significant way.