spacetime. In these theories ordinary distances in space and ordi-nary time intervals are not true scalars, but the magnitude of the spacetime displacement is a scalar. The magnitude of the spacetime displacement depends on the properties of the spacetime, and these properties are determined by what is called a “metric.” However, for many purposes the approximation that distance and time are sepa-rately scalars is a good one. In fact, the approximation is good when-ever Newtonian mechanics is a good approximation, and Newtonian mechanics provides an excellent description of the motion of ordi-nary macroscopic objects on the earth. We discuss Newton’s laws of motion in Chapter 5 and his theory of gravity in Chapter 6. So, until we discuss relativity, we shall treat distance and time separately as scalars.
3.5 Velocity and acceleration
A number of scalars and vectors are important in the study of mo-tion. We have already discussed several scalars (temperature, dis-tance, and time) and vectors (displacement and force). We next con-sider some of these quantities in more detail and introduce some oth-ers.
Suppose a traveler can drive to Boston from New York, a distance, say, of 350 kilometers (km) in 5 hours (hr). Then, knowing both the distance and the time, we can define a derived quantity, the average speed, which is the distance traveled divided by the time required to go that distance. In our example, the average speed is 70 kilometers
per hour (70 km/hr), which is obtained by dividing 350 km by 5 hr.
A speed of 70 km/hr is about 44 miles per hour (mi/hr).
Let us introduce the symbol v for “speed” and ¯v for “average speed.” (Our notation here is that a line on top of a symbol denotes its average value. In later chapters we use the line with a different meaning.) Our definition of average speed of a body is that it is the distance traveled, which we denote by the symbol s, divided by the elapsed time, which we denote by the symbol t. Thus, using our sym-bols, we can write ¯v = s/t. By using symsym-bols, we not only shorten our notation but can manipulate the symbols algebraically. How-ever, we shall rarely use algebra in this book. Unfortunately, there is a price to pay for using symbols: we either have to remember what the symbols mean or write down their definitions.
If our driver looks at his speedometer from time to time during the trip from New York to Boston, he will notice that the speedometer does not always read 70 km/hr, but sometimes more and sometimes less. In order to know how fast the car is going at any given time, we need the concept of speed at a given instant of time. But an instant of time is no time at all. If speed is distance divided by time, and the time is zero, how can we define the speed? We learned quite early in arithmetic that we are not allowed to divide by zero.
We obtain a solution to the problem as follows: We divide the to-tal distance into a number of smaller distance segments, and measure the time interval required to travel each segment. Then the average speed during each segment can be calculated by dividing the seg-ment by the time interval to traverse it. As we make the distance segments smaller and smaller, the corresponding time intervals to
3.5. VELOCITY AND ACCELERATION 39 traverse each of them will also become smaller. When the distance segments and time intervals are so small that the car cannot apprecia-bly change its speed during a single interval, then the average speed in the interval is approximately the same as the speed throughout the interval. Thus, we can define the speed at a given time as the average speed during a very small time interval which includes the given time. A more precise definition is that the speed is the limit of the average speed as both the distance segment and time interval become arbitrarily small. The branch of mathematics that is suited to calculate with quantities that become arbitrarily small is called the calculus. We do not discuss the calculus in detail in this book.
Just as we define the scalar speed as a distance segment divided by the corresponding time interval, so we can define a vector velocity as a displacement segment divided by the corresponding time in-terval. The vector velocity is often denoted by the symbol v. The scalar speed v is the magnitude of the velocity. Unfortunately, not even physicists are always precise in their language, and sometimes use the word “velocity” when they mean “speed.” Because of this imprecision, one is often forced to infer the precise meaning from context.
We previously mentioned a car changing its speed as it travels along. The car also changes its direction from time to time. When-ever a body changes its speed, its direction, or both, it is changing its velocity. If the velocity of an object changes, we say that it undergoes acceleration, usually denoted by the symbol a. Like velocity, accelera-tion is a vector, although the magnitude of the acceleraaccelera-tion, a scalar, is also called acceleration. Because we use the same word for vector
and scalar acceleration, we have to make the meaning clear by con-text. The symbols, however, are different, being a for the vector and afor the scalar.
We can define the acceleration as the change in velocity during a very short time interval divided by the time interval, or, in other words, acceleration is the rate of change of velocity. In the case of a car speeding up or slowing down on a straight road, the car acceler-ates by virtue of a change in speed. In the case of a car going around a curve at constant speed, the car accelerates by virtue of a change in direction. Of course, a car can change its speed while going around a curve, and this too is acceleration. In ordinary language, slowing down is sometimes called “deceleration,” but we often use the same word acceleration for slowing down, speeding up, or changing di-rection with or without a change in speed.
Many people confuse the concepts of velocity and acceleration.
Because velocity and acceleration are vectors, each has a direction, and these directions may be different. If the velocity and acceleration of an object are in the same direction, the object moves in a straight line and speeds up. If the acceleration is in the opposite direction to the velocity, the object moves in a straight line and slows down. If the direction of the acceleration makes an angle with the direction of the velocity, the object moves in a curved path.
It is possible for the velocity of an object to be zero at a certain instant, while at that same instant the acceleration is not zero. For example, if you throw a ball straight up in the air, then, when the ball is at its highest point above the ground, the velocity of the ball is zero. Its acceleration at that point is not zero, however, because even
3.5. VELOCITY AND ACCELERATION 41 though the velocity is momentarily zero, the velocity is changing. If the velocity and acceleration were both zero, the velocity would not change, and the ball would remain motionless in the air, clearly in conflict with observation.
Velocity and acceleration have different units. Because velocity is a displacement divided by a time, the unit of velocity is a unit of displacement, say, meters (m) divided by a unit of time, say, seconds (s), or m/s. The unit of acceleration is the unit of velocity (m/s) divided by the unit of time (s), or m/s/s. This unit is also written m/s2.
Chapter 4
Early Ideas of Motion
Paradox. An assertion that is essentially self-contradictory, although based on a valid deduction from acceptable premises.
—The American Heritage Dictionary (Houghton Mifflin, 1985)