ACCELLERATION

1. ACCELLERATION • -Acceleration shows how fast velocity changes • -Acceleration is the “velocity of velocity”

2. Uniform acceleration Uniform motion v(t) x(t) v Δx v0 t t t0 Δt t v(t) Δx t Δt

3. For any motion: For uniform acceleration: Acceleration Velocity Uniform acceleration Uniform motion

4. Base equations for 1D uniform acceleration 2 equations 2 quantities can be found • -What to remember? • -How to use? Two useful equations that can be derived from the base equations See how it was derived on previous slide

5. Example:A train car moves along a long straight track. The graph shows the position as a function of time for this train. The graph shows that the train: 1. speeds up all the time. 2. slows down all the time. 3. speeds up part of the time and slows down part of the time. 4. moves at a constant velocity. Steepness of slope is decreasing time Positive Acceleration = a smile Negative Acceleration = a frown  position position  time time

6. t1 t0 Example: The graph shows position as a function of time for two trains running on parallel tracks. Which of the following is true? 1. At time t0, both trains have the same velocity. 2. Both trains speed up all the time. 3. Both trains have the same velocity at some time beforet0. 4. Somewhere on the graph, both trains have the same acceleration. Same slope at t = t1

7. v = slope of x(t) a = slope of v(t) or a = curvature of x(t) Position Velocity Acceleration

8. Displacement = area under v(t) curve Change in velocity = area under a(t) curve t1 t0 Position Velocity Acceleration

9. Example v(t) from a(t): Draw the velocity vs. time graph that corresponds to the following acceleration vs. time graph. Assume that the velocity at t = 0 is zero. a v v v t t t t Does your graph look like one of these? A C B

10. v a t t NB: a < 0 but object is speeding up. NB: a > 0 but object is slowing up.

11. Free fall -Free fall acceleration: g=9.8m/s2 Using the two base equations: Substitute the following into the base equations: To derive the following equations:

12. Example 1.A particle, a material point, is thrown vertically up. Find the maximum height the particle will reach and the time it will take, if you are given the initial height and the initial velocity. Given: Unknown variables: Solution: Answer:

13. Example 2. A particle, a material point, is thrown vertically up. Find the velocity with which the particle returns to the point from which it was thrown, and the time this flight will take. The initial height and the initial velocity are given. Given Solution: Answer Compare to example 1:

14. Example 3, Two particles, material points, are thrown vertically up. One particle is thrown before the other. Find the time at which both objects are at the same height, and the height at which the objects’ intersection occurs. Given: Equations used: Unknown variables: Too many variables but Thus, 3 equations and 3 unknowns. Solution: Answer:

15. Free fall (review) Example1: Ball #1 is thrown vertically upwards with a speed of v0 from the top of a building and hits the ground with speed v1. Ball #2 is thrown vertically downwards from the same place with the same speed v0 and hits the ground with speed v2. Which one of the following three statements is true. Neglect air resistance. A. v1>v2 B. v1=v2 C. v1<v2 D. Depends on which ball is more massive E. None of the above Example2: You are throwing a ball straight up in the air. At the highest point, the ball’s 1. velocity and acceleration are zero. 2. velocity is nonzero but its acceleration is zero. 3. acceleration is nonzero, but its velocity is zero. 4. velocity and acceleration are both nonzero.