Unit 1b: Motion in One Dimension-Constant Acceleration Test Date : Thursday September 21th

Equation of motion for constant-velocity linear motion

When velocity is not constant On a velocity-versus-time graph, velocity will be a straight horizontal line only if it is constant. Instantaneous velocity is the velocity of an object at a particular time. Average velocity is the ratio of the change in position and the time interval during which this change occurred. For motion at constant velocity, the instantaneous and average velocities are equal; for motion with changing velocity, they are not.

Relationship Between Acceleration & Velocity Constant velocity (shown by red arrows maintaining the same size) Acceleration equals zero

Relationship Between Velocity & Acceleration Acceleration and velocity are in SAME directions AKA SPEEDING UP!!! Velocity and acceleration are in the same direction Acceleration is constant (blue arrows maintain the same length) Velocity is increasing (red arrows are getting longer) Positive velocity and positive acceleration

Relationship Between Velocity & Acceleration Acceleration and velocity are in OPPOSITE directions AKA SLOWING DOWN!!! Acceleration and velocity are in opposite directions Acceleration is constant (blue arrows maintain the same length) Velocity is decreasing (red arrows are getting shorter) Velocity is positive and acceleration is negative

Acceleration is the rate at which the velocity of an object is changing When the velocity of the object increases or decreases by the same amount during the same time interval (a constant rate of change) is constant acceleration. In this simple case, the velocity-versus-time graph will be linear.

RECALL Equation for the Velocity time graph using y = kx +b Y, the depended variable, is the Velocity (V) K, the slope, is the acceleration (a) X, the independent variable, is time (t) B, the intercept, is the starting velocity of the object.

Finding acceleration from a velocity-versus-time graph Acceleration is the slope of the velocity-versus-time graph: The units of the slope are m/s/s or m/s2 A larger slope indicates the velocity is increasing at a faster rate. Velocity is a vector quantity; therefore acceleration is also a vector quantity. The average acceleration of an object during a time interval is

Acceleration on a v vs t graph! Acceleration can be positive or negative. If an object moving in the positive direction: Which is speeding up and which is slowing down?!

Acceleration on a v vs t graph! Acceleration can be positive or negative. If an object moving in the negative direction: Which is speeding up and which is slowing down?!

Graphing Constant Velocity OR No Acceleration x t v t a t

Constant Acceleration on Position vs. Time and Velocity vs. Time Graphs Graphing Constant Acceleration Distance-time graph Velocity-time graph Position (m) 10 20 30 1 3 2 5 4 Time (s) Velocity (m/s) -10 10 20 1 3 2 5 4 Time (s) -20 Distance traveled is changing Change in velocity is constant

a = 2 m/s2 + direction - direction 4 16 12 8 60 24 20 56 52 48 44 40 16 12 8 60 24 20 56 52 48 44 40 36 32 28 64 68

- direction a = 2 m/s2 + direction = acceleration = velocity 4 16 12 8 16 12 8 60 24 20 56 52 48 44 40 36 32 28 64 68 = acceleration = velocity

a = 2 m/s2 + direction - direction = acceleration = velocity 4 16 12 8 16 12 8 60 24 20 56 52 48 44 40 36 32 28 64 68 = acceleration = velocity Time (s) Position (m) 10 30 20 40 60 50 1 3 2 4 6 5 Position vs Time Time (s) Velocity (m/s) 4 12 8 16 24 20 1 3 2 6 5 Velocity vs Time Time (s) Acceleration (m/s2) 1 3 2 4 6 5 Acceleration vs Time

Why the position vs. time graph is quadratic with acceleration © 2014 Pearson Education, Inc.

Why the position vs. time graph is quadratic with acceleration Position is quadratic in time (there is a t2 term), so the graph is parabolic. The slopes of the tangent lines (indicating the instantaneous velocity) are different for different times.

How acceleration affects finding displacement in a velocity time graph © 2014 Pearson Education, Inc.

How acceleration affects finding displacement in a velocity time graph The equation for displacement can be found from the area between the velocity-versus-time graph line and the time axis. © 2014 Pearson Education, Inc.

Acceleration Graphs The area under an acceleration vs. time graph = the change in velocity (Δv) Acceleration tells you how fast you speed up, slow down, or change direction.

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Three equations of motion 𝑉 𝑥 2 = 𝑉 0 2 +2 𝑎 𝑥 (𝑋− 𝑋 0 ) © 2014 Pearson Education, Inc.

Super Secret Equation Number 4

REMEMBER YOU PICK DIRECTION AND ZERO! All motion is going in a direction you assign! So if you assign right as positive or negative, everything in that direction has to be what you assigned.

This car is traveling LEFT because the Velocity is POSITIVE! + - Kinematics Example #1 A car is traveling 40 m/s and has a constant acceleration of -4.75 m/s2 for 6 seconds. How fast is the car traveling at the end? -4.75 = (vx - 40) 6 = Δ t 40 m/s -4.75 m/s2 ? 6 s = a = v◦ = vx (6 ) (6) Δ v = vx - v◦ -28.5 = vx - 40 + 40 40 + 11.5 m/s= vx This car is traveling LEFT because the Velocity is POSITIVE!

This car is traveling RIGHT because the Velocity is POSITIVE! + - Kinematics Example #2 A car traveling with an initial velocity of 6 m/s to the right, accelerates at 2 m/s2, for 6 seconds. What is the car’s final velocity? ? 6 m/s 2 m/s2 6 s = vx Vx = V◦ + aΔt = v◦ Vx = 6 + (2)(6) = a = Δt Vx = 18 m/s This car is traveling RIGHT because the Velocity is POSITIVE!

This car is STILL traveling RIGHT because the Velocity is NEGATIVE! + - Kinematics Example #2 A car traveling with an initial velocity of 6 m/s to the right, accelerates at 2 m/s2, for 6 seconds. What is the car’s final velocity? ? -6 m/s -2 m/s2 6 s = vx Vx = V◦ + aΔt = v◦ Vx = -6 + (-2)(6) = a = Δt Vx = -18 m/s This car is STILL traveling RIGHT because the Velocity is NEGATIVE!

Vf = 20 m/s Δx= ½ (vx + v◦)Δt 65 = ½ (vx+ 6) (2) (2 ) 130 = (vx + 6) 5 Kinematics Example #3 A car traveling with an initial velocity of 6 m/s accelerates for 5 seconds and travels a distance of 65m. What is the car’s final velocity? Δx= ½ (vx + v◦)Δt ? 6 m/s 5 s 65 m = vx 65 = ½ (vx+ 6) (2) (2 ) = v◦ 130 = (vx + 6) 5 = Δt 5 = Δx 26 = (vx + 6) Vf = 20 m/s -6 -6

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Mathematics of free fall The acceleration of objects in free fall is given a special symbol g: We replace x by y in our equations of motion for constant acceleration because free fall is placed along the vertical axis. These equations use the choice that the positive y-axis is downward! © 2014 Pearson Education, Inc.

Motion diagrams and graphs for free fall © 2014 Pearson Education, Inc.

Earth’s gravity causes objects to accelerate downward Vertical Motion Earth’s gravity causes objects to accelerate downward On a very large body (like a planet) all objects accelerate downward at the same rate On Earth, objects accelerate downward at: 9.8 m/s2 Other planets have different acceleration rates, but they are always constant for that planet on their surfaces. Ex. Mars—3.7 m/s2

Earth’s gravity causes objects to accelerate downward Vertical Motion + - + - OR Earth’s gravity causes objects to accelerate downward If an object at rest falls and drops 10 m, how fast will it be moving just before it hits the ground? All motion directed downward, so if you assign down as positive or negative, everything in that direction has to be what you assigned

Vertical Motion (Down Only) + - Earth’s gravity causes objects to accelerate downward If an object at rest falls and drops 10 m, how fast will it be moving just before it hits the ground? 0 m/s 10 m ? m/s

Vertical Motion (Up Only) + - Earth’s gravity causes objects to accelerate downward If an object is thrown upwards 10 m, how long will it take to reach it’s highest point?