# V t p t x 8 sin4 t given 8 cos4 t given y a t p t x

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v(t) =p¤(t) =X-8sin(4t)given,8cos(4t)givenYa(t) =p¤¤(t) =X-322cos(4t)given,-322sin(4t)givenYFor the physical interpretation, recall the classic physicsequation, “Force=mass×acceleration.” A force actingon a mass induces acceleration (i.e., the mass moves);acceleration acting on a mass induces a force (gravitygives our mass aweight). Thus force and accelerationare closely related. A moving ball “wants” to travel ina straight line. Why does the ball in our example movein a circle? It is attached to your hand by a string. Thestring applies a force to the ball, affecting it’s motion:the stringacceleratesthe ball. This is not accelerationin the sense of “it travels faster;” rather, this accelerationischanging the velocityof the ball. In what direction isthis force/acceleration being applied? In the direction ofthe string, towards your hand.Hence it makes sensea(t)is parallel top(t), but has adifferent magnitude and points in the opposite direction.The magnitude of the acceleration is related to the speedat which the ball is traveling. A ball whirling quicklyis rapidly changing direction/velocity. When velocity ischanging rapidly, the acceleration must belarge .Question 164.An object moves in a spiral with position func-tionp(t) =OEcost,sint, tº,where distances are measured in meters and time is in minutes.Describe the object’s velocity and acceleration at timet.v(t) =(- sint,cost,1)a(t) =(- cost,- sint,0).Question 165.What is the speed of this object?v(t) =ø2m_ minQuestion 166.What is the angle betweenvanda?The angle equals_2radiansFeedback(correct):Since the speed is constant, the velocity andthe acceleration are perpendicular.Projectile motionAn important application of vector-valued position functions isprojectile motion: the motion of objects under the influence ofgravity. We will measure time in seconds, and distances willeither be in meters or feet. We will show that we can completelydescribe the path of such an object knowing its initial positionand initial velocity (where itisand where itis going.)Suppose an object has initial positionp(0) =OEx(0), y(0)º300
Motion and paths in spaceand initial velocity ofv(0) =v0OEcos( ),sin( )º.Here,is often called theangle of elevation. Since the accel-eration of the object is known, namelya(t) =OE0,-gº,wheregis the gravitational constant, we can findp(t)knowingour two initial conditions. We first findv(t):v(t) =˚a(t)dtv(t) =˚OE0,-gºdtv(t) =OE0,-gtº+v(0).We integrate once more to findp(t):p(t) =˚v(t)dtp(t) =˚OE0,-gtº+C1dtp(t) =@0,-gt22A+tv(0) +p(0).You can adjust the initial position,P0, angle, magnitude of thevelocity, and magnitude of the acceleration below:Geogebra link:We demonstrate how to solve for a position function in thecontext of projectile motion in the next example.

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Term
Fall
Professor
Buenger
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