Motion with Constant Acceleration

1. Motion with Constant Acceleration McNutt - Physics Binder For Notes You need:

2. Objectives • Describe motion in terms of frame of reference, displacement, time intervaland velocity. • Calculate displacement, average velocity, time intervaland acceleration. • Draw and interpret position vs. time and velocity vs. time graphs.

3. The Story so far…. • The average velocity for any motion is • Where Δx is the displacement and Δt is the time interval. • The instantaneous velocity v is the velocity the object has at a particular time. • It is the average velocity over a very short time interval.

4. Position vs. Time for Constant Velocity Motion • If the velocity is constant, the instantaneous velocity is the average velocity. • v= vAV • The graph is a straight line. • The position is given by the equation

5. Position vs. Time for Accelerated Motion • Here the average velocity is not constant. • For the instantaneous velocity, take the average velocity over a very short time interval. • Graphically, this is the slope of the tangent line of the graph.

6. Acceleration • When velocity changes, we have an acceleration. • Velocity can change in magnitude or direction. • Average acceleration is given by the formula:

7. v a v a v a v a v a = 0 v = 0 a or Accelerations can be positive or negative in 1-d motion.

8. Constant Acceleration Model • Accelerations can vary with time. • Many situations in physics can be modeled by a constant acceleration. • Constant acceleration means the object changes velocity at a constant rate. • When dealing with a constant acceleration situation, we will drop the subscript “AV”.

9. Velocity vs. time for constant acceleration • aAV is the slope of the velocity vs. time graph. • If the velocity vs. time graph is a straight line, the acceleration is constant. • In this case, the formula for velocity is v (m/s) t (s)

10. Displacement on a Velocity vs. Time graph • Since and v is the height of the area under the velocity versus time graph, and t is the base of the velocity versus time graph, the area under a velocity versus time graph shows the displacement. Δx

11. Displacement for constant acceleration • The displacement from time 0 to time t is the area under the velocity graph from 0 to t. • Area = ½ b h v (m/s) t (s)

12. Displacement for constant acceleration • If the initial velocity is not zero, we have to include a rectangular piece. • Triangle Area = ½ b h • Rectangle = l x w v (m/s) t (s)

13. Displacement for constant acceleration • If we don’t know vf, we can calculate it from a. • Area =l w + ½ b h v (m/s) t (s)

14. Equations of Motion for Constant Acceleration • Now we have derived three equations that apply to the motion with constant acceleration model

15. Formulas for other time intervals • If the motion begins at some other time other than t = 0, then we simply replace t with the time interval Δt.

16. Practice Problems • #1- An automobile with an initial speed of 4.3 m/s accelerates uniformly at the rate of 3.0 m/s2. Find the final speed and the displacement after 5.0 s. Remember to list the GIVENS & UNKOWNS when setting up your equations! v t Constant Acceleration

17. Practice Problem Continued v t

18. Practice Problems • #2 - A car starts from rest and travels for 5.0 s with a uniform acceleration of -1.5 m/s2. What is the final velocity of the car? How far does the car travel in this time interval?

19. One Other Equation for Constant Acceleration • All of the equations we have so far for this model involve time. • Sometimes, we are not told the time over which the motion occurs. • We can use two of these equations to eliminate time.

20. Equations for the Constant Acceleration Model

21. Practice Problems • A jet plane lands with a speed of 100 m/s and can accelerate uniformly at a maximum rate of -5.0 m/s2 as it comes to rest. Can this airplane land at an airport where the runway is 0.80 km long?

22. Practice Problems • #3 Constant Acceleration