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Lecture 04: Kinematics + Dynamics Kinematics Equations constant acceleration Dynamics Newton’s Second Law Non-zero acceleration Equations for Constant Acceleration x v = x0 + v0t + ½ at2 = v0 + at v2 = v02 + 2a(x-x0) x is final position xo is initial position v is final velocity v0 is initial velocity a is acceleration t is time Kinematics Example A car is traveling 30 m/s and applies its breaks to stop. Assuming constant acceleration of -6 m/s2, how long does it take for the car to stop, and how far does it travel before stopping? x = x0 + v0t + ½ at2 v = v0 + at v2 = v02 + 2a(x-x0) Begin by using the second equation to find the time: v = v0 + a t Then use the first equation to find the distance: (0 m/s) = (30 m/s) + (-6 m/s2) t t=5s x = x0 + v0t + ½ a t2 x = (0 m) + (30 m/s)(5s) + ½ (-6 m/s2) (5 s)2 x = 75 m Dynamics: F = ma We have already dealt with situations where a = 0. But when the net force is not zero, there IS an acceleration! Dynamics Example A tractor is pulling a trailer with a constant acceleration. If the forward acceleration is 1.5 m/s2, Calculate the force on the trailer (m = 400 kg) due to the tractor. (Consider just the trailer.) FBD: x-direction: FN FT = ma FT Fg F = ma FT = 600 N y-direction: F = ma FN - Fg = 0 y x FN = 3920 N FT = 600 N Summary of Concepts • Constant Acceleration x = x0 + v0t + ½ at2 v = v0 + at v2 = v02 + 2a(x-x0) • F = m a – Draw Free Body Diagram – Write down equations – Solve Kinematics Example A car moving at 15 m/s is traveling toward an intersection and sees the light turn yellow. The car accelerates at 4 m/s2 until it gets to the intersection 18 m away. How long does it take the car to get to the intersection? (And assuming the light is yellow for 1 s, does the car make it before the light turns red?) Note: even after accelerating, the car is still traveling safely under the speed limit… There are two ways to solve this…I will use one method and you will use the other! Kinematics Example A car moving at 15 m/s is traveling toward an intersection and sees the light turn yellow. The car accelerates at 4 m/s2 until it gets to the intersection 18 m away. How long does it take the car to get to the intersection? (And assuming the light is yellow for 1 s, does the car make it before the light turns red?) Note: even after accelerating, the car is still traveling safely under the speed limit… I will do this in two steps: First, using v2 = v02 + 2 a (x-x0), we can solve for the final velocity. Second, using v = v0 + a t, we can solve for the time. Kinematics Example A car moving at 15 m/s is traveling toward an intersection and sees the light turn yellow. The car accelerates at 4 m/s2 until it gets to the intersection 18 m away. How long does it take the car to get to the intersection? (And assuming the light is yellow for 1 s, does the car make it before the light turns red?) Note: even after accelerating, the car is still traveling safely under the speed limit… v2 = v02 + 2 a (x-x0) v2 = (15 m/s)2 + 2 (4 m/s2) (18 m – 0 m) v = 19.2 m/s v = v0 + a t (19.2 m/s) = (15 m/s) + (4 m/s2) t t = 1.05 s The car does not make it before the light turns red… It should have just stopped!