Chapter 3: Acceleration and Accelerated Motion Unit 3 Accelerated Motion

Chapter 3: Acceleration and Accelerated Motion t (time-seconds)v (velocity-m/s)x (position-m)a (acceleration-m/s 2 ) Velocity/Time Graph velocity (m/s) t (seconds) The slope means acceleration! The slope means something!

Chapter 3: Acceleration and Accelerated Motion What equation can we get from this graph? The constant acceleration equation! We can also get the “how fast” equation. From Graph: From Algebra: Chapter 3: Acceleration and Accelerated Motion

t (time-seconds)v (velocity-m/s)x (position-m)a (acceleration-m/s 2 ) Chapter 3: Acceleration and Accelerated Motion acceleration (m/s2) t (seconds) What would the acceleration/time graph look like? Horizontal line means constant acceleration. Let’s look at the area under the ‘curve.’ 5s 10 m/s 2 It is the change in velocity!

Chapter 3: Acceleration and Accelerated Motion t (time-seconds)v (velocity-m/s)x (position-m)a (acceleration-m/s 2 ) velocity (m/s) t (seconds) Area of a triangle: 10 m/s 1 s How do you find displacement from a velocity/time graph? Area under the ‘curve.’ 5

Chapter 3: Acceleration and Accelerated Motion t (time-seconds)v (velocity-m/s)x (position-m)a (acceleration-m/s 2 ) velocity (m/s) t (seconds) Area of a triangle: 20 m/s 2 s 20

Chapter 3: Acceleration and Accelerated Motion t (time-seconds)v (velocity-m/s)x (position-m)a (acceleration-m/s 2 ) velocity (m/s) t (seconds) Area of a triangle: 30 m/s 3 s 45

Chapter 3: Acceleration and Accelerated Motion t (time-seconds)v (velocity-m/s)x (position-m)a (acceleration-m/s 2 ) velocity (m/s) t (seconds) Area of a triangle: 40 m/s 4 s 80

Chapter 3: Acceleration and Accelerated Motion t (time-seconds)v (velocity-m/s)x (position-m)a (acceleration-m/s 2 ) velocity (m/s) t (seconds) Area of a triangle: 50 m/s 5 s What is happening to the amount of distance increased after each second? 125

Chapter 3: Acceleration and Accelerated Motion t (time-seconds)v (velocity-m/s)x (position-m)a (acceleration-m/s 2 ) Chapter 3: Acceleration and Accelerated Motion Shape: Top opening parabola (curvy up) What is the proportionality? This is the “How Far” equation! (Starting with zero velocity)

Chapter 3: Acceleration and Accelerated Motion t (time-sec)t ^2 (time^2)-sec^2)X (position-meters) It checks out!!

Chapter 3: Acceleration and Accelerated Motion Let’s look at another way to get the “How Far” equation. From the velocity time graph: velocity (m/s) t (seconds) Area under curve = displacement From previous “How Fast” equation: flip

Chapter 3: Acceleration and Accelerated Motion On a position time graph what is slope equal to? Velocity Is the slope constant in this graph? No You can use a tangent line to tell you the slope at a given point in time. Let’s try. Finding the slope at 3 seconds: Draw a tangent line, which is a straight line that touches the curve at only the desired point. This is instantaneous velocity. (The velocity at that instant.) Find Average Velocity

Chapter 3: Acceleration and Accelerated Motion Can we make a motion map of this motion? You Bet! x (displacement s1s2s3s4s 5s vvvvv What happens to the distance between the dots? What is happening to the velocity?

Chapter 3: Acceleration and Accelerated Motion We need to make a change one addition to our “How Far” equation. What if you saw a velocity/time graph that looked like this? velocity (m/s) t (seconds) What is different about this graph than the previous velocity/time graph? The velocity at t = 0 is 10 m/s. In other words, the car has an initial velocity of 10 m/s.

Chapter 3: Acceleration and Accelerated Motion velocity (m/s) t (seconds) Let’s see how this affects our “how far” equation. Again, we need to find displacement. How do we do this? Area under ‘curve’ Let’s look at the time interval of 0 – 1 sec. This area is a goofy, irregular shape, so we need to look at this as a rectangle and a triangle together! Green Area What equation can I make for the area (displacement)? Red Area

Chapter 3: Acceleration and Accelerated Motion velocity (m/s) t (seconds) Green AreaRed Area Let’s do the same thing for 0 – 2 sec. Look for the pattern: time initial velocity Therefore the ‘How Far’ equation becomes:

Chapter 3: Acceleration and Accelerated Motion Now use that equation to find the position of the object at each second. t (time-seconds)v (velocity-m/s)x (position-m)a (acceleration-m/s 2 ) This comes from slope, which is the same as the first v/t graph. At 3 sec From previous pages At 4 sec. At 5 sec.

Chapter 3: Acceleration and Accelerated Motion t (time-seconds)v (velocity-m/s)x (position-m)a (acceleration-m/s 2 ) Let’s make a position/time graph for this motion. Notice the shape: top opening parabola (curvy up) How can the position time graph go through (0,0) and the velocity time graph didn’t? The car can have an initial velocity at t=0, at the ref. point.

Chapter 3: Acceleration and Accelerated Motion 0s1s2s3s4s vvvvv x (displacement Let’s make a motion map for this motion also. t (time-seconds)v (velocity-m/s)x (position-m)a (acceleration-m/s 2 ) s

Chapter 3: Acceleration and Accelerated Motion Let’s take the case of an object slowing down…(Negative acceleration) t (time- seconds) v (velocity-m/s)x (position-m)a (acceleration-m/s 2 ) velocity (m/s) t (seconds) Calculate the slope: What does slope of a v/t graph mean again??? Oh yeah….. Acceleration! What does negative acceleration mean?

Chapter 3: Acceleration and Accelerated Motion What does negative acceleration mean? It can mean slowing down, but that’s not a complete picture. It most accurately means that the object is accelerating in the negative direction. Ex: If your put your car in reverse at the stop sign (reference pt.) and put your foot on the gas pedal, you would be speeding up in the backwards direction. This would also be negative acceleration. VelocityAccelerationMotion Positive Negative Positive Speeding up, forward Slowing down forward Speeding up, backward Slowing down, backward

Chapter 3: Acceleration and Accelerated Motion acceleration (m/s 2 ) 0 t (seconds) t (time-seconds)v (velocity-m/s)x (position-m)a (acceleration-m/s 2 )

Chapter 3: Acceleration and Accelerated Motion velocity (m/s) t (seconds) Let’s make the position/time graph: x (position-meters) time (t-seconds) Find  x at t=1 Let’s use the “how far” equation.

Chapter 3: Acceleration and Accelerated Motion