2.2 Acceleration
- 2. Objectives The student will be able to: • Describe motion in terms of changing velocity • Compare graphical representations of accelerated and non-accelerated motions • Apply kinematic equations to calculate distance, time, or velocity under conditions of constant acceleration
- 3. Acceleration Acceleration: the rate of change of velocity • Units: meters per second per second or meters per second squared (m/s2) • Symbol: a
- 4. Average Acceleration Equation 푎 = Δ푣 Δ푡 = 푣푓 − 푣푖 푡푓 − 푡푖 average acceleration = change in velocity time required for change • Acceleration has both direction and magnitude.
- 5. Sample Problem 2B – p.49 As a shuttle bus comes to a normal stop, it slows down from 9.00 m/s to 0.00 m/s in 5.00s. Find the average acceleration of the bus.
- 6. Problem Tip Watch for implied data in problem statements, such as “starts at rest” (vi = 0 m/s) or “comes to rest” (vf = 0 m/s)
- 7. Practice 2B Problems p.49 #1, 3-5
- 8. The signs of the velocity and acceleration combine to describe an object’s motion. 풗풊 a Motion + + speeding up _ _ speeding up + _ slowing down _ + slowing down - or + 0 constant velocity 0 - or + speeding up from rest 0 0 remaining at rest
- 9. Graphing Acceleration On a velocity versus time graph, the slope of the line connecting one point and the next indicates the average acceleration. slope = change in vertical change in horizontal 푎푎푣푔 = Δ푣 Δ푡
- 10. Acceleration Graph Describe the acceleration at points A, B, C.
- 11. Think/Pair/Share Conceptual Challenge on p.50
- 12. Constant Acceleration What is it? Demo - Walking
- 13. Deriving Displacement w/Constant Acceleration What we know so far.. 푣푎푣푔 = Δ푥 Δ푡 For objects moving with constant acceleration 푣푎푣푔 = 푣푖+푣푓 2 avg. velocity = 푖푛푖푡푖푎푙 푣푒푙표푐푖푡푦+푓푖푛푎푙 푣푒푙표푐푖푡푦 2
- 14. Deriving Displacement w/Constant Acceleration Δ푥 Δ푡 = 푣푎푣푔= 푣푖 + 푣푓 2
- 15. Displacement w/Constant Acceleration Δ푥 = 1 2 (푣푖 + 푣푓 )Δ푡 displacement = 1 2 (initial velocity + final velocity)(time interval) • You will know that acceleration is constant by the phrase “uniform negative/positive acceleration”
- 16. Sample Problem 2C– p.53 A race car reaches a speed of 42 m/s. It then begins a uniform negative acceleration, using its parachute and braking system, and comes to rest 5.5 s later. Find out how far the car moves while stopping.
- 17. Practice 2C Problems p.53 #1, 3-5
- 18. Velocity with Constant Acceleration What if we don’t know the vf but we still want to calculate displacement… 푎푎푣푔 = Δ푣 Δ푡 = 푣푓−푣푖 Δ푡
- 19. Velocity with Constant Acceleration 푣푓 = 푣푖 + 푎Δ푡 final velocity = initial velocity + (acceleration · time interval)
- 20. Displacement w/Constant Acceleration We can combine the previous equations: Δ푥 = 1 2 (푣푖 + 푣푓 )Δ푡 and 푣푓 = 푣푖 + 푎Δ푡 to form Δ푥 = 1 2 (푣푖 + 푣푖 + 푎Δ푡)Δ푡 Δ푥 = 1 2 (2푣푖Δ푡 + 푎Δ푡2)
- 21. Displacement w/Constant Acceleration Δ푥 = 푣푖Δ푡 + 1 2 푎Δ푡2 displacement = (initial velocity · time interval) + 1 2 acceleration · (time interval)2
- 22. Sample Problem 2D – p.55 A plane starting at rest at one end of a runway undergoes a uniform acceleration of 4.8 m/s2 for 15 s before takeoff. What is its speed at takeoff? How long must the runway be for the plane to be able to take off?
- 23. Practice 2D Problems p.55 #1-3
- 24. So far all equations have required knowing the time interval. If we don’t know t, we need to rearrange an equation and use substitution.
- 25. Rearrange for Δ푡: Δ푥 = 1 2 (푣푖 + 푣푓 )Δ푡
- 26. Then substitute into: 푣푓 = 푣푖 + 푎Δ푡
- 27. Final Velocity After any Displacement 푣푓 2 = 푣푖 2 + 2푎Δ푥 • When you use this equation, you must take the square root of the equation to find the vf. • The square root can be either positive or negative, you will determine which value is right by reasoning based on the direction of motion.
- 28. Sample Problem 2E – p.57 A person pushing a stroller starts from rest, uniformly accelerating at a rate of 0.500 m/s2. What is the velocity of the stroller after it has traveled 4.75 m?
- 29. Practice 2E Problems p.58 #1,3,5