Graphical Interpretation of Differential Equations in Rectilinear Motion

Graphical Interpretations Interpretation of the differential equations governing rectilinear motion is considerably clarified by representing the relationships among s v a and t graphically Figure 23a is a schematic plot of the variation of s with t from time t1 to time t2 for some given rectilinear motion By constructing the tangent to the curve at any time t we obtain the slope which is the velocity v dsdt Thus the velocity can be determined at all points on the curve and plotted against the corresponding time as shown in Fig 23b Similarly the slope dvdt of the vt curve at any instant gives the acceleration at that instant and the at curve can therefore be plotted as in Fig 23c We now see from Fig 23b that the area under the vt curve during time dt is v dt which from Eq 21 is the displacement ds Consequently the net displacement of the particle during the interval from t1 to t2 is the corresponding area under the curve which is s2s1 ds t2t1 v dt or s2 s1 area under vt curve Similarly from Fig 23e we see that the area under the at curve during time dt is a dt which from the first of Eqs 22 is du Thus the net change in velocity between t1 and t2 is the corresponding area under the curve which is v2v1 dv t2t1 a dt or v2 v1 area under at curve Note two additional graphical relations When the acceleration a is plotted as a function of the position coordinate s Fig 24a the area under the curve during a displacement ds is a ds which from Eq 23 is v dv du dv22 Thus the net area under the curve between position coordinates s1 and s2 is v2v1 dv s2s1 a ds or 12 v2 v12 area under as curve When the velocity v is plotted as a function of the position coordinate s Fig 24b the slope of the curve at any point A is dvds By constructing the normal AB to the curve at this point we see from the similar triangles that CBlB dvds Thus from Eq 23 CB dv a the acceleration It is necessary that the velocity and position coordinate axes have the same numerical scales so that the acceleration read on the position coordinate scale in meters or feet say will represent the actual acceleration in meters or feet per second squared The graphical representations described are useful not only in visualizing the relationships among the several motion variables but also in obtaining approximate results by graphical integration or differentiation The latter case occurs when a lack of knowledge of the mathematical relationship prevents its expression as an explicit mathematical function which can be integrated or differentiated Experimental data and motions which involve discontinuous relationships between the variables are frequently analyzed graphically KEY CONCEPTS Analytical Integration If the position coordinate s is known for all values of the time t then successive mathematical or graphical differentiation with respect to t gives the velocity v and acceleration a In many problems however the functional relationship between position coordinate and time is unknown and we must determine it by successive integration from the acceleration Acceleration is determined by the forces which act on moving bodies and is computed from the equations of kinetics discussed in subsequent chapters Depending on the nature of the forces the acceleration may be specified as a function of time velocity or position coordinate or as a combined function of these quantities The procedure for integrating the differential equation in each case is indicated as follows a Constant Acceleration When a is constant the first of Eqs 22 and 23 can be integrated directly For simplicity with s s0 v v0 and t 0 designated at the beginning of the interval then for a time interval t the integrated equations become v v0 dv a 0t dt or v v0 at s s0 dv a s s0 ds or v2 v02 2as s0 Substitution of the integrated expression for v into Eq 21 and integration with respect to t give s s0 ds t 0 v0 at dt or s s0 v0t 12 at2 These relations are necessarily restricted to the special case where the acceleration is constant The integration limits depend on the initial and final conditions which for a given problem may be different from those used here It may be more convenient for instance to begin the integration at some specified time t1 rather than at time t 0 Caution The foregoing equations have been integrated for constant acceleration only A common mistake is to use these equations for problems involving variable acceleration where they do not apply b Acceleration Given as a Function of Time a ft Substitution of the function into the first of Eqs 22 gives ft dvdt Multiplying by dt separates the variables and permits integration Thus v v0 dv t 0 ft dt or v v0 t 0 ft dt Velocity and Acceleration The average velocity of the particle during the interval Δt is the displacement divided by the time interval or vavg ΔsΔt As Δt becomes smaller and approaches zero in the limit the average velocity approaches the instantaneous velocity of the particle which is v limΔt0 ΔsΔt or v dsdt 21 Thus the velocity is the time rate of change of the position coordinate s The velocity is positive or negative depending on whether the corresponding displacement is positive or negative The average acceleration of the particle during the interval Δt is the change in its velocity divided by the time interval or aavg ΔvΔt As Δt becomes smaller and approaches zero in the limit the average acceleration approaches the instantaneous acceleration of the particle which is a limΔt0 ΔvΔt or a d²sdt² ddt 22 The acceleration is positive or negative depending on whether the velocity is increasing or decreasing Note that the acceleration would be positive if the particle had a negative velocity which was becoming less negative If the particle is slowing down the particle is said to be decelerating Velocity and acceleration are actually vector quantities as we will see for curvilinear motion beginning with Art 23 For rectilinear motion in the present article where the direction of the motion is that of the given straightline path the sense of the vector along the path is described by a plus or minus sign In our treatment of curvilinear motion we will account for the changes in direction of the velocity and acceleration vectors as well as their changes in magnitude By eliminating the time dt between Eq 21 and the first of Eqs 22 we obtain a differential equation relating displacement velocity and acceleration This equation is v dv a ds or dsdt ds ds 23 Equations 21 22 and 23 are the differential equations for the rectilinear motion of a particle Problems in rectilinear motion involving finite changes in the motion variables are solved by integration of these basic differential relations The position coordinate s the velocity v and the acceleration a are algebraic quantities so that their signs positive or negative must be carefully observed Note that the positive directions for v and a are the same as the positive direction for s From this integrated expression for v as a function of t the position coordinate s is obtained by integrating Eq 21 which in form would be s s₀ ds t 0 v dt or s s₀ t 0 v dt If the indefinite integral is employed the end conditions are used to establish the constants of integration The results are identical to those obtained by using the definite integral If desired the displacement s can be obtained by a direct solution of the secondorder differential equation s ft obtained by substitution of ft into the second of Eqs 22 SAMPLE PROBLEM 21 The position coordinate of a particle which is confined to move along a straight line is given by s 2t³ 24t 6 where s is measured in meters from a convenient origin and t is in seconds Determine a the time required for the particle to reach a velocity of 72 ms from its initial condition at t 0 b the acceleration of the particle when v 30 ms and c the net displacement of the particle during the interval from t 1 s to t 4 s SAMPLE PROBLEM 22 A particle moves along the xaxis with an initial velocity vx 50 ftsec at the origin when t 0 For the first 4 seconds it has no acceleration and thereafter it is acted on by a retarding force which gives it a constant acceleration ax 10 ftsec² Calculate the velocity and the xcoordinate of the particle for the conditions of t 8 sec and t 12 sec and find the maximum positive xcoordinate reached by the particle The springmounted slider moves in the horizontal guide with negligible friction and has a velocity v0 in the sdirection as it crosses the midposition where s 0 and t 0 The two springs together exert a retarding force to the motion of the slider which gives it an acceleration proportional to the displacement but oppositely directed and equal to a ks² where k is constant The constant is arbitrarily squared for later convenience in the form of the expressions Determine the expressions for the displacement s and velocity v as functions of the time t A freighter is moving at a speed of 8 knots when its engines are suddenly stopped If it takes 10 minutes for the freighter to reduce its speed to 4 knots determine and plot the distance s in nautical miles moved by the ship and its speed v in knots as functions of the time t during this interval The deceleration of the ship is proportional to the square of its speed so that a kv²