Worksheet 1-Using the Derivative to Solve Motion Problems 1. Analyzing the Motion A 2 kg object's position as a function of time is given by the equation: x(t)=2t33t + t +1 15 10 5 A = 0 Where t is given in seconds and x is measured in meters. Find the following (with units): B = -5 v(t) = a(t) x(t) = -10 x(1) = x(3) = = v(t) = 15 v(1) = v(3) = = a(t) = -15 0 1 s 2 s a(1) = a(3) = For the motion that occurred between t = 1s and t = 3 s, find the average values: = = 3. Two-dimensional motion The position as a function of time for a humming bird is given by the equations: The impulse and change in momentum are given by J = FAt x(t) = 2t+1 y(t) = 3t 2 - Ap = mvf-mvi According to Newton, the impulse causes the change in momentum, and they should be equal mathematically. For the motion that occurred between t = 1s and t = 3 s, calculate the impulse and change in momentum. Impulse = Change in momentum = Are they equal? At what time will the speed of this object equal 7 m/s? (Recall, the speed is defined to be the magnitude of the velocity vector.) Where is the humming bird at this time? What is its acceleration at this time? Time = Position Acceleration 2. Curve fitting the motion to stop a car You are driving your car at 10 m/s. At time t = 0, you stop the car, and it takes you 2 s to come to a complete stop. This motion is given by the equation: x(t) = A sin(Bt) 0