AccelerationGraphs to FunctionsChanging VelocityAverage AccelerationInstantaneous AccelerationSecond DerivativeConstant AccelerationAcceleration and PositionAcceleration RelationshipsTake OffAccelerationAccelerationGraphs to FunctionsGraphs to FunctionsA simple graph of constant A simple graph of constant velocity corresponds to a velocity corresponds to a position graph that is a position graph that is a straight line.straight line.The functional form of the The functional form of the position is position is This is a straight line and This is a straight line and only applies to straight lines.only applies to straight lines.xtvtv0x000xtvx Changing VelocityChanging VelocityIn more complicated motion the velocity is not In more complicated motion the velocity is not constant.constant.We can express a time rate of change for velocity just We can express a time rate of change for velocity just as for position, as for position, vv = = vv22 - - vv11..The The accelerationacceleration is the time rate of change of is the time rate of change of velocity: velocity: aa = = vv / / t.t.Average AccelerationAverage AccelerationExample problemExample problemA jet plane has a takeoff speed of 250 km/h. If the A jet plane has a takeoff speed of 250 km/h. If the plane starts from rest, and lifts off in 1.2 min what is plane starts from rest, and lifts off in 1.2 min what is the average acceleration?the average acceleration?aa = = vv / / tt = [(250 km/h) / (1.2 min)] * (60 min/h) = [(250 km/h) / (1.2 min)] * (60 min/h)aa = 1.25 x 10 = 1.25 x 1044 km/h km/h22Why is this so large? Is it reasonable?Why is this so large? Is it reasonable?Does the jet accelerate for an hour?Does the jet accelerate for an hour?Instantaneous Instantaneous AccelerationAccelerationInstantaneous velocity is Instantaneous velocity is defined by a derivative.defined by a derivative.Instantaneous acceleration Instantaneous acceleration is also defined by a is also defined by a derivative.derivative.dtdxtxvt 0limdtdvtvat 0limvtP1P2P3P4Second DerivativeSecond DerivativeThe acceleration is the derivative of velocity with The acceleration is the derivative of velocity with respect to time.respect to time.The velocity is the derivative of position with respect The velocity is the derivative of position with respect to time.to time.This makes the acceleration the second derivative of This makes the acceleration the second derivative of position with respect to time. position with respect to time. 22)(dtxddtdxdtddtdva Constant AccelerationConstant AccelerationConstant velocity gives a Constant velocity gives a straight line position graph.straight line position graph.Constant acceleration gives Constant acceleration gives a straight line velocity graph.a straight line velocity graph.The functional form of the The functional form of the velocity is velocity is vtata0v000vtav Acceleration and PositionAcceleration and PositionFor constant acceleration the For constant acceleration the average acceleration equals average acceleration equals the instantaneous the instantaneous acceleration.acceleration.Since the average of a line Since the average of a line of constant slope is the of constant slope is the midpoint:midpoint:002000021)21( xtvtaxtvtax vtv0½ta0(½t) + v0Acceleration RelationshipsAcceleration RelationshipsAlgebra can be used to Algebra can be used to eliminate time from the eliminate time from the equation.equation.This gives a relation This gives a relation between acceleration, between acceleration, velocity and position.velocity and position.For an initial or final velocity For an initial or final velocity of zero. This becomesof zero. This becomes•xx = = vv22 / 2 / 2aa•vv22 = 2 = 2 a xa x 00221xtvatx avvt0   0002000020221xavvvavvxxavvvavvax0vatv fromTake OfTake OfExample problemExample problemA jet plane has a takeoff speed of 250 km/h. If the A jet plane has a takeoff speed of 250 km/h. If the plane starts from rest, and has a constant plane starts from rest, and has a constant acceleration of 1.25 x 10acceleration of 1.25 x 1044 km/h km/h22, what is the length of , what is the length of the runway?the runway?xx = = vv22 / 2 / 2aa = (250 km/h) = (250 km/h)22 / (2.5 x 10 / (2.5 x 1044 km/h km/h22))xx = 2.5 km = 2.5 kmIs this reasonable?Is this