10/8/2002, VELOCITY Math 21a, O. KnillCURVES. A vector-valued map on the real line r(t) = (x(t), y(t)) or r(t) = (x(t), y(t), z(t)) is called a curve.EXAMPLES.1) r(t) = (cos(t), sin(t)) is a circle in the plane.2) r(t) = (cos(t), sin(t), t) is a spiral in space.3) r(t) = P + tv = (P1+ tv1, P2+ tv2, P3+ tv3) is a straight line connecting P (t = 0) with Q (t = 1).DERIVATIVES. If r(t) = (x(t), y(t), z(t)) is a vector valued function describing a curve, thenr0(t) = (x0(t), y0(t), z0(t)) = ( ˙x, ˙y, ˙z)is called the derivative of r. (The notation with the dot is common, when the parameter is time.) Thederivative is also called the velocity. The length of the velocity vector is called the speed. The derivativeof the velocity is called acceleration. While the velocity vector is tangent to the curve, the acceleration canpoint in any direction.EXAMPLE. If r(t) = (cos(3t), sin(2t), 2 sin(t)), then r0(t) = (−3 sin(3t), 2 cos(2t), 2 cos(t)).WHAT IS MOTION?The paradoxon of Zeno of Elea: ”If we look ata body at a specific time, then the body is fixed.Having it fixed at each time, there is no motion”.While one might wonder today a bit about Zeno’snaivity, there were philosophers in our time likeKant, Hume or Hegel, who thought about Zeno’sparadoxons. Also physisists continue to ponderabout the question what is time and space.WHAT IS A DERIVATIVE?The derivative or rate of change is a limit. It can be approximated by the vector (r(t + dt) −r(t))/dt, where dtis a small number. If dt approaches zero, and the limit exists, the velocity exists at this point. If r(t) = P + vtis a line, then r0(t) = v.EXAMPLES.1) If r(t) = P + vt is a line, then r0(t) = v.2) If r(t) = (|t|, t2,√t + 1), then r0(t) = (sign(t), 2t, 2t/√t + 1). The derivative exists at all times except att = 0 and t = −1.EXAMPLES OF VELOCITIES.Electrons in Metals: 0.005 m/sPerson walking: 1.5 m/sCar: 15-50 m/sSignals in nerves: 40 m/sAeroplane: 70-900 m/sSound in air: Mach1=340 m/sSatellite: 1200 m/sSpeed of bullet: 1200-1500 m/sEarth around the sun: 30’000 m/sSun around galaxy center: 200’000 m/sLight in vacuum: 300’000’000 m/sEXAMPLES OF ACCELERATIONS.Train: 0.1-0.3 m/s2Car: 3-8 m/s2Space shuttle: ≤ 3G = 30m/s2Combat plane (F16) (blackout): 9G=90 m/s2Ejection from F16: 14G=140 m/s2.Free fall: 1G = 9.81 m/s2Electron in vaccum tube: 1015m/s2INTEGRATION. If v(t) = (x(t), y(t), z(t)) is a curve, thenRt0v(t) dt is defined as(Rt0x(t) dt,Rt0y(t) dt,Rt0z(t) dt).APPLICATION. A flight recorder in a space object records the accelerations (a(t), b(t), c(t) = (sin(2t), sin(t), t)in x, y, z direction. The accelerations are accessible because they are proportional to forces, the device canmeasure. If the plane is at rest at (0, 0, 0) when t = 0, where is it at t = 10π?ANSWER. We know r00(t) = (cos(t), sin(t), t). By integration, we obtainRt0r00(t) dt = r0(t) =(cos(t)/2, −cos(t), t2/2) and r(t) = (−sin(2t)/4, −sin(t), t3/6). At t = 10π, we have r(10π) = (0, 0, 1000π3/6).ARC LENGTH. If r(t) is a curve defined on some parameter interval [a, b], and v(t) = r0(t) is the velocity and||v(t)|| is the speed, thenRba||v(t)|| dt is called the length of the curve.Written out, the formula isL =Zbapx0(t)2+ y0(t)2+ z0(t)2dt .EXAMPLE. Let r(t) = (cos(t), sin(t), t) be a spiral defined for t ∈ [0, 10π] then ||v(t)|| =√2 and the length is10π√2.REMARK. Often we will not be able to find a closed formula for the lenght of a curve. For example the Lissajouxfigure r(t) = (cos(3t), sin(5t)) has lengthR2π0q9 sin2(3t) + 25 cos2(5t) dt which can be evaluated numericallyonly.NEWTONS LAW.Newton second law saysm¨t = mr00(t) = F (t)where F (t) is the external force acting onthe body and m is the mass of the body.GRAVITY. If F (t) = (0, 0, −gm), thena body feels a constant acceleration to-wards the ground:¨r = −g = −9.81m/s2.QUESTION. If we drop a body from height h, how long does it take to hit the ground?ANSWER. The position at time t is (0, 0, h − gt2/2). For t =p2h/g, we are at the ground. For example, ifh = 10 meters, then we have wait about 1.4 seconds.PROBLEM. In the movie ”Six days, seven nights” (with HarrisonFord), pirats shoot with a cannon onto the plane of the heros.They aim however vertically up onto the plane. The 25mm bullethas an initial speed of 35m/s. How much time do the pirats haveuntil the boat is hit by their own bullet? Assume g = 10.ANSWER. If v(t) is the speed of the cannon shell, then v0(t) = −gand v(t) = 35 −10 ∗t. The velocity is zero after 3.5 seconds. Thepirats have therefore 7 seconds to leave the boat.HISTORY OF NEWTON’S LAW.Ancient Greek philosophers thought that the motions of the stars and planets were unrelated to events on theearth. The understanding of gravity changed with Galileo, Kepler, Brahe and Newton in the 16’th century.Galileo realized that the gravitational acceleration is independent of the mass of the body. By 1666 Newton didnot understand the mechanics of circular motion yet. In 1666 he imagined that the Earth’s gravity is influencedthe Moon, counterbalancing its centrifugal force. From his law of centrifugal force and Kepler’s third law ofplanetary motion, Newton deduced the inverse-square law. In 1679 Newton corresponded with Hooke who hadwritten to Newton claiming ”... that the attraction always is in a duplicate proportion to the distance from thecenter reciprocall”. But Newton then himself derived Kepler’s laws from the law of central forces. (See Book:”Huygens and Barrow, Newton and Hooke” by V.I. Arnold) No portrait survives of Robert Hooke. His name issomewhat obscure today, due in part to the enmity of his famous, influential, and extremely vindictive collegueSir Isaac
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