2. LINEAR MOTION You will study the motion of an object in one dimension from a number of points of view. You will demonstrate how the variables of motion are related by differentiation and integration and investigate the relationship between potential and kinetic energy. Theory Why Study Motion? Motion is everywhere in the universe. Only at a temperature of absolute zero is the motion in any body truly absent. If motion exists then so also does energy. To the delight of the physicist the tools that were invented by Galileo, Newton and others 200 years ago to describe motion apply everywhere in the known universe, from electrons in our own bodies to the farthest galaxy. The study of motion and of energy is at the heart of physics. This experiment deals with the simplest kind of motion, that in one dimension. Kinematics and Dynamics The subject of motion is divided into kinematics and dynamics. Kinematics is concerned with the aspects of motion which exclude the forces which cause motion; thus, in a manner of speaking, kinematics is focussed on the development of definitions: position, displacement, velocity, acceleration and on the relationships between them. Dynamics widens the study of motion to include force and energy. Definitions Position meaning the object is to the right of the reference Kinematics begins with the idea of position. Sup- point. We could also agree that, by convention, the pose that we photograph an object moving to the sign of x is to be positive here. left along a horizontal path at two instants of time and superimpose the images for study (Figure 1). Elapsed Time We examine one image with a ruler and mark off The two positions of the object in Figure 1 must be the number of units which separates the object labelled with different vectors and different clock from the ruler’s zero. The zero is a reference or times. The photographs can be said to show two origin at a position of zero units by definition while events, an initial “i” event and a final “f” event. the object is at another position, say x units. x is an There is now an elapsed time between the events instantaneous quantity since it applies to a specific given by: clock time—the instant of the taking of the photo- graph. Position like length is a basic quantity being Dt = t f – t i , …[1] dependent only on the unit adopted. But position involves also direction: in prin- (unit seconds, abbreviated s). Keep in mind the ciple the object could be to our right or to our left. difference between the two concepts of time; an To include the information of direction we use a elapsed time is the difference between two clock vector. The magnitude or length of the vector, say times. r, is x (or r), while the direction is to the right, B2-1 2. Linear Motion displacement 0 rf clock time tf clock time ti D r = r f – r i object object r i D r = v D t Figure 1. Shown here is an object moving toward the origin (left) at two positions with the corresponding clock times indicated. Position, displacement and velocity vectors are given different head styles to emphasize their different nature. Displacement entials d. Eq[3] then becomes what is known as the Displacement differs from position. In the interval instantaneous velocity of time between the events the object moves from drr one position to another. The displacement is the = vr . …[4] difference between the two vectors describing the dt two positions This quantity is abstract and tricky to imagine: it r r r Dr = rf – ri , …[2] can be thought of as the average velocity that might be measured (with a superior detection sys- (unit meters, abbreviated m). Displacement, being tem) over an infinitely short elapsed time (or the the difference between two vectors, is a vector. The velocity at a specific clock time). In practice, it can displacement is negative (according to our conven- be obtained only approximately. tion) since it points towards the origin. If the displacement is known as an analytical function of time, r(t), then the instantaneous vel- Velocity ocity at some clock time t0 is the tangent to the Another quantity in kinematics is the average vel- function at t0, or the first derivative of r(t) at t0. The ocity, or the displacement an object undergoes in finding of tangents is one of the objectives of this one second of elapsed time. This is the ratio experiment. r r r Dr rf – ri r Acceleration = = v , …[3] Dt Dt The velocity of the object in Figure 1 may change with time. The velocity could decrease as a result (unit meters per second, abbreviated m.s –1). The of a force of friction with the path. Or the velocity average velocity, being a vector divided by a could increase if the path were not horizontal and scalar, is a vector. The average velocity is negative a component of the force of gravity acts on the here, too, since it points towards the origin; its object. The time rate of change of the average magnitude is the speed. velocity is called the average acceleration and the The elapsed time in eqs[1] and [3] is a finite in- time rate of change of the instantaneous velocity is terval. What if it is infinitely small? Mathemati- called the instantaneous acceleration. Both types of cally, this amounts to taking the limit of eq[3] as acceleration are defined as in eqs[3] and [4] with Dt® 0. The increments D are replaced by the differ- “v” subsituted for “r “and “a” substituted for “v”. B2-2 Linear Motion 2 Motion of an Object Whose Velocity is Constant In this experiment you will mostly be comcerned The computer software used in this experiment with the motion of an object whose velocity is does something similar by finding the average changing. However, for purposes of understand- velocity by averaging over the slopes between a ing we first consider the case of constant velocity. number of datapairs (7 by default). Thus if a num- The case of an object moving towards the ori- ber of datapoints were measured and the results gin on a horizontal plane is drawn in Figure 2. We plotted on a graph, the result might resemble suppose that the data pairs (t, r), where t is the Figure 4. clock time and r is the position are measurable at As the glider approaches the origin here the regular intervals by some detection system. Two position decreases but is always positive; while the such points when plotted on a graph might appear velocity remains at a constant negative value. The as shown in the upper half of Figure 3. velocity is therefore just the derivative or the slope of A computer could be programmed to calculate the displacement versus clock time graph (or the the “average velocity” as the slope between the slope of the position versus clock time graph here two datapoints and plot it as a point on a graph in one dimension). The velocity is seen to change (lower half of Figure 3). The result is negative, the little (if at all) with clock time and so the accelera- sign indicating the direction of the velocity vector. tion (decceleration) is very small. Motion Detector clock time: tf clock time: ti ri 0 rf positive displacement v = D r also to the left D r = rf – ri D t Figure 2. An object is shown at two positions (events) while moving toward a detector on a horizontal plane. ( ti , ri ) ( tf , rf ) ( tf , vf ) Figure 3. A graph of the two position-clock time datapoints described in Figure 2. Shown also is a point on the velocity graph as it might be generated from the slope between the two datapoints multiplied by the sign of the velocity vector. B2-3 2 Linear Motion Figure 4. Typical position and velocity graphs as might be produced for an object moving as shown in Figure 2. Motion of an Object Whose Velocity Changes with Time In this experiment you will mostly be ignoring the velocity decrease. (The object is slowing down.) effects of the force of friction. However, for The velocity and acceleration versus clock time purposes of understanding it is useful to consider graphs in this case will resemble Figure 5. friction, however briefly. It is known from other experiments (“Simple We know that a small force of friction exists Measurements”) that the force of friction, though between the glider and the layer of air on which it small, has a complicated functional form giving moves. Friction acts opposite to the direction of rise to a decceleration which depends on the first motion (to the right in Figure 2) and therefore pro- (and sometimes the second) power of the velocity. duces an acceleration also toward the right. This Gravity, unlike friction, is a constant force and is acceleration is often described as a decceleration in therefore much easier to deal with; the effect of that it is opposite to the velocity and describes a gravity on motion we consider in the next section. Figure 5. Velocity and acceleration graphs for an object moving as shown in Figure 2 while subject to a small force of friction. Remember, graphed here are the magnitudes of the vectors multiplied by the sign corresponding to the direction of the vectors. Motion of an Object Whose Acceleration is Constant and Nonzero Figure 6 shows an object moving with constant Starting from the assumption that the accelera- non-zero acceleration down an inclined plane tion is constant and nonzero we can develop the (neglecting friction).