Describing Motion: Kinematics in One Dimension 1

Describing Motion: Kinematics in One Dimension “Point” Particles & Large Masses Translational Motion = Straight line motion. Rotational Motion = Moving (rotating) in a circle. Oscillations = Moving (vibrating) back & forth in same path. Continuous Media Waves, Sound, Light Fluids = Liquids & Gases Conservation Laws: Energy Momentum Angular Momentum Just Newton’s Laws expressed in other forms! 2

Describing Motion: Kinematics in One Dimension Unit 3 Topics: 3A: Reference Frames & Displacement 3B: Average Velocity 3C: Instantaneous Velocity 3D: Acceleration 3E: Motion at Constant Acceleration 3F: Solving Problems 3G: Freely Falling Objects 3

Terminology Mechanics = Study of objects in motion. 2 parts to mechanics. Kinematics = Description of HOW objects move. Dynamics = WHY objects move. Introduction of the concept of FORCE. Causes of motion, Newton’s Laws 4

Terminology Translational Motion = Motion with no rotation. Rectilinear Motion = Motion in a straight line path. 5

Sect. 3A: Reference Frames Every measurement must be made with respect to a reference frame. Usually, speed is relative to the Earth. 6

Example 3A-1: Reference Frames If you are sitting on a train & someone walks down the aisle, the person’s speed with respect to the train is a few km/hr, at most. The person’s speed with respect to the ground is much higher. Specifically, if a person walks towards the front of a train at 5 km/h (with respect to the train floor) & the train is moving 80 km/h with respect to the ground. What is the person’s speed, relative to the ground? 85 km/h. 7

Sect. 3A: Reference Frames When specifying speed, always specify the frame of reference unless its obvious (“with respect to the Earth”). Distances are also measured in a reference frame. When specifying velocity or distance, we also need to specify DIRECTION. 8

Sect. 3A: Reference Frames – Coordinate Axes Define reference frame using a standard coordinate axes. 2 Dimensions (x,y) Note, if its convenient, we could reverse + & - ! - ,+ +,+ - , - + , - Standard set of x,y coordinate axes 9

Sect. 3A: Reference Frames – Coordinate Axes 3 Dimensions (x,y,z) Define direction using these. First Octant 10

Sect. 3A: Reference Frames – Displacement & Distance Distance traveled by an object ≠ Displacement of the object! 11

Sect. 3A: Reference Frames – Displacement & Distance Displacement = change in position of object. Displacement is a vector (magnitude & direction). Distance is a scalar (magnitude). Example: distance = ?, displacement = ? 100 m; 40 m East 12

Sect. 3A: Reference Frames – Displacement x1 = 10 m, x2 = 30 m Displacement ≡ ∆x = x2 - x1 = 20 m ∆ ≡ Greek letter “delta” meaning “change in” tI ↓ tF ↓ ← times The arrow represents the displacement (in meters). 13

Example 3A-2: Displacement xI = 30 m, xF = 10 m Displacement ≡ ∆x = xF - xI = - 20 m Displacement is a VECTOR 14

Sect. 3A: Reference Frames – Vectors & Scalars Many quantities in physics, like displacement, have a magnitude and a direction. Such quantities are called VECTORS. Other quantities which are vectors: velocity, acceleration, force, momentum, ... Many quantities in physics, like distance, have a magnitude only. Such quantities are called SCALARS. Other quantities which are scalars: speed, temperature, mass, volume, ... 15

Sect. 3A: Reference Frames – Vectors & Scalars Some texts use BOLD letters to denote vectors. In others, vectors are denoted with arrows over the symbol, In one dimension, we can drop the arrow and remember that a + sign means the vector points to right & a minus sign means the vector points to left. V 16

Sect. 3B: Average Velocity Average Speed ≡ (Distance traveled)/(Time taken) Average Velocity ≡ (Displacement)/(Time taken) Velocity: Both magnitude & direction Describes how fast an object is moving. A VECTOR. (Similar to displacement). Speed: Magnitude only Describing how fast an object is moving. A SCALAR. (Similar to distance). Units: distance/time = m/s Scalar→ Vector→ 17

Sect. 3B: Average Velocity, Average Speed Displacement from before, walk for 70 s. Average Speed = ( 100 m)/(70 s) = 1.4 m/s Average velocity = ( 40 m)/(70 s) = 0.57 m/s 18

Sect. 3B: Calculating Average Velocity In general: ∆x = xF - xI = displacement ∆t = tF - tI = elapsed time Average Velocity: = (xF - xI)/(tF - tI) tI ↓ tF ↓ ← times V = Bar denotes average 19

Example 3B-1: Calculating Average Velocity A person runs from xI = 50.0 m to xF = 30.5 m in ∆t = 3.0 s. ∆x = ?; V = ? Average velocity = V = (∆x)/(∆t) = -(19.5 m)/(3.0 s) = -6.5 m/s. Negative sign indicates DIRECTION, (negative x direction) 20

Sect. 3C: Instantaneous Velocity Instantaneous velocity ≡ velocity at any instant of time ≡ average velocity over an infinitesimally short time Mathematically, instantaneous velocity: ≡ ratio considered as a whole for smaller & smaller ∆t. Mathematicians call this a derivative. **Do not set ∆t = 0 because ∆x = 0 then & 0/0 is undefined! ⇒ Instantaneous velocity = V 21

instantaneous velocity = average velocity ↓ These graphs show (a) constant velocity → and (b) varying velocity → instantaneous velocity ≠ average velocity 22

Sect. 3C: Instantaneous Velocity The instantaneous velocity is the average velocity in the limit as the time interval becomes infinitesimally short. Ideally, a speedometer would measure instantaneous velocity; in fact, it measures average velocity, but over a very short time interval. Figure 2-8. Caption: Car speedometer showing mi/h in white, and km/h in orange. 23

Velocity can change with time. Sect. 3D: Acceleration Velocity can change with time. An object with velocity that is changing with time is said to be accelerating. Definition: Average acceleration ≡ ratio of change in velocity to elapsed time. a ≡ = (vF - vI)/(tF - tI) Units: velocity/time = distance/(time)2 = m/s2 Acceleration is a vector. 24

Sect. 3D: Instantaneous Acceleration 25

Example 3D-1: Average Acceleration A car accelerates along a straight road from rest to 90 km/h in 5.0 s. Find the magnitude of its average acceleration. 26

Example 3D-1: Avg Acceleration - Ans A car accelerates along a straight road from rest to 90 km/h in 5.0 s. Find the magnitude of its average acceleration. Note: 90 km/h = 25 m/s a = = (25 m/s – 0 m/s)/5 s = 5 m/s2 27

Example 3D-2: Conceptual Question If velocity & acceleration are both vectors, then Are the velocity and the acceleration always in the same direction? 28

Example 3D-2: Conceptual Question Ans Are the velocity and the acceleration always in the same direction? NO!! If the object is slowing down, the acceleration vector is in the opposite direction of the velocity vector! 29

Example 3D-3: Car Slowing Down A car moves to the right on a straight highway (positive x-axis). The driver puts on the brakes. If the initial velocity (when the driver hits the brakes) is 15.0 m/s and it takes 5.0 s to slow down to 5.0 m/s. Calculate the car’s average acceleration. 30

Example 3D-3: Car Slowing Down A car moves to the right on a straight highway (positive x-axis). The driver puts on the brakes. If the initial velocity (when the driver hits the brakes) is 15.0 m/s and it takes 5.0 s to slow down to 5.0 m/s. Calculate the car’s average acceleration. a = = (v2 – v1)/(t2 – t1) = (5 m/s – 15 m/s)/(5s – 0s) a = - 2.0 m/s2 31

Example 3D-4: Acceleration/Deceleration If the same car is moving to the left instead of to the right (still assume positive x is to the right). The car is decelerating & the initial & final velocities are the same as before. Calculate the average acceleration now. 32

Example 3D-4: Acceleration/Deceleration Ans The same car is moving to the left instead of to the right. Still assume positive x is to the right. The car is decelerating & the initial & final velocities are the same as before. “Deceleration”: A word which means “slowing down” and avoided in physics. Instead (in one dimension), positive & negative acceleration are used. This is because (for one dimensional motion), deceleration does not necessarily mean the acceleration is negative. 33

Example 3D-5: Conceptual Question If velocity & acceleration are vectors, then is it possible for an object to have a zero acceleration and a non-zero velocity? 34

Example 3D-5: Conceptual Question Ans If velocity & acceleration are vectors, then is it possible for an object to have a zero acceleration and a non-zero velocity? YES!! If the object is instantaneously at rest (v = 0) but is either on the verge of starting to move or is turning around & changing direction, the velocity is zero, but the acceleration is not! 35