Chapter 2: Motion, Forces, & Newton’s Laws

Note! I think that, logically, the topics in Chs. 2 & 3 should be covered in a slightly different order than your book does them. So, I will not cover them in the same order as the book. Further, I will treat Chs. 2 & 3 as if they were only one chapter!

Brief Overview of the Course “Point” Particles & Large Masses Translational Motion = Straight line motion. Chapters 2,3,4,6,7 Rotational Motion = Moving (rotating) in a circle. Chapters 5,8 THE COURSE THEME IS NEWTON’S LAWS OF MOTION!! Oscillations = Moving (vibrating) back & forth on the same path. Chapter 11

Brief Overview of the Course Continuous Media Waves & Sound Chapters 11,12 Fluids = Liquids & Gases Chapter 10 THE COURSE THEME IS NEWTON’S LAWS OF MOTION!! Conservation Laws Energy, Momentum, Angular Momentum These are Newton’s Laws expressed in other forms!

Some Chapter 2 Topics Reference Frames & Displacement Average Velocity Instantaneous Velocity Acceleration Motion at Constant Acceleration Solving Problems Freely Falling Objects

Description of HOW objects move. Terminology Mechanics = Study of objects in motion. 2 parts to mechanics. Kinematics = Description of HOW objects move. Chapters 2 & 3 Dynamics = WHY objects move. Introduction of the concept of FORCE. Causes of motion, Newton’s Laws Most of the course from Chapter 4 & beyond. For a while, Assume Ideal Point Masses (no physical size). Later, extended objects with size.

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

Reference Frames & Displacement Every measurement must be made with respect to a reference frame. Usually, the speed is relative to the Earth.

Reference Frames & Displacement Every measurement must be made with respect to a reference frame. Usually, the speed is relative to the Earth. For example, 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.

Reference Frames & Displacement Every measurement must be made with respect to a reference frame. Usually, the speed is relative to the Earth. For example, 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. The person’s speed, relative to the ground is 85 km/h.

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

(Cartesian or rectangular) Coordinate Axes Usually, we define a reference frame using a standard coordinate axes. (But the choice of reference frame is arbitrary & up to us, as we’ll see later!) 2 Dimensions (x,y) Note, if it is convenient, we could reverse + & - ! - ,+ +,+ + , - A standard set of xy (Cartesian or rectangular) coordinate axes - , -

Coordinate Axes 3 Dimensions (x,y,z) First Octant Define direction using these. First Octant

Displacement & Distance Distance traveled by an object  Displacement of the object! Here, Distance = 100 m. Displacement = 40 m East. Displacement  Δx  Change in position of an object. Δx is a vector (magnitude & direction). Distance is a scalar (magnitude).

Displacement  ∆x = x2 - x1 = 20 m  t2   times x1 = 10 m, x2 = 30 m Displacement  ∆x = x2 - x1 = 20 m ∆  Greek letter “delta” meaning “change in” The arrow represents the displacement (meters).

Displacement  ∆x = x2 - x1 = - 20 m x1 = 30 m, x2 = 10 m Displacement  ∆x = x2 - x1 = - 20 m Displacement is a VECTOR

Vectors and 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, ...

Units of both are distance/time = m/s Average Velocity Average Speed  (Distance traveled)/(Time taken) A Vector A Scalar Average Velocity  (Displacement)/(Time taken) Velocity: Both magnitude & direction describing how fast an object is moving. It is a VECTOR. Speed: Magnitude only describing how fast an object is moving. It is a SCALAR. Units of both are distance/time = m/s

Average Velocity & Average Speed Consider the displacement from before. Suppose that the person does the whole trip in 70 s. Average Speed = (100 m)/(70 s) = 1.4 m/s Average Velocity = (40 m)/(70 s) = 0.57 m/s

Average Velocity vave = (x2 - x1)/(t2 - t1) General Case  t2   times vave = ∆x = x2 - x1 = displacement ∆t = t2 - t1 = elapsed time Average Velocity vave = (x2 - x1)/(t2 - t1)

The Average Velocity is the slope of the line Velocity and Position Consider the case where the position vs. time curve is as shown in the figure. In general, The Average Velocity is the slope of the line segment that connects the positions at the beginning & end of the time interval.

Example 2.1: Velocity of a Bicycle Calculate the average velocity from t = 2.0 s to 3.0 s. The displacement is Δx = 12 m – 5 m = 7 m. So vave = (Δx)/(Δt) = (7 m)/(1 s) = 7 m/s

Instantaneous Velocity Average velocity doesn’t tell us anything about details during the time interval. To look at some of the details, smaller time intervals are needed The slope of the curve at the time of interest will give the instantaneous velocity at that time.

Instantaneous Velocity Instantaneous Velocity  The velocity at any instant of time  The Average Velocity over an infinitesimally short time. Mathematically, the Instantaneous Velocity is formally defined as:  the ratio considered as a whole for infinitesimally small ∆t. Mathematicians call this a “derivative”. Do not set ∆t = 0 because ∆x = 0 then & 0/0 is undefined!  Instantaneous velocity v

Instantaneous Velocity  Velocity  Instantaneous Velocity at any instant of time. Mathematically, instantaneous velocity: Mathematicians call this a “derivative”.  Instantaneous Velocity v ≡ Time Derivative of Displacement x

(a) Constant Velocity  These graphs show (a) Constant Velocity  Instantaneous Velocity = Average Velocity and (b) Varying Velocity  Instantaneous Velocity  Average 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.

Acceleration An object’s velocity can change with time. An object with a velocity that is changing with time is said to be accelerating. Definition: Average acceleration = ratio of change in velocity to elapsed time. aave  = (v2 - v1)/(t2 - t1) Acceleration is a vector. Instantaneous acceleration Units: velocity/time = distance/(time)2 = m/s2

Graphical Analysis of Velocity (Example 2.3) To find the velocity graphically: Find the slope of the line tangent to the x-t graph at the appropriate times For the average velocity for a time interval, find the slope of the line connecting the two times

Example: 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. Note: 90 km/h = 25 m/s aave =

Example: 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. Note: 90 km/h = 25 m/s aave = = (25 m/s – 0 m/s)/5s = 5 m/s2

Are the velocity and the acceleration always in the same direction? Conceptual Question Velocity & Acceleration are both vectors. Are the velocity and the acceleration always in the same direction?

Are the velocity and the acceleration always in the same direction? Conceptual Question Velocity & Acceleration are both vectors. 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!

Example: Car Slowing Down A car moves to the right on a straight highway (positive x-axis). The driver puts on the brakes. The initial velocity (when driver hits brakes) is v1 = 15.0 m/s. It takes t = 5.0 s to slow down to v2 = 5.0 m/s. Calculate the car’s average acceleration.

Example: Car Slowing Down A car moves to the right on a straight highway (positive x-axis). The driver puts on the brakes. The initial velocity (when driver hits brakes) is v1 = 15.0 m/s. It takes t = 5.0 s to slow down to v2 = 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

Deceleration 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 same as before. Calculate the average acceleration now.

positive & negative acceleration. Deceleration “Deceleration” is a word which means “slowing down”. We try to avoid using it in physics. Instead (in one dimension), we talk about positive & negative acceleration. This is because (for one dimensional motion) deceleration does not necessarily mean the acceleration is negative!

Is it possible for an object to have a zero Conceptual Question Velocity & Acceleration are both vectors. Is it possible for an object to have a zero acceleration and a non-zero velocity?

YES!! Conceptual Question Is it possible for an object to have a zero Velocity & Acceleration are both vectors. Is it possible for an object to have a zero acceleration and a non-zero velocity? YES!! If the object is moving at a constant velocity, the acceleration vector is zero!

Velocity & acceleration are both vectors. Conceptual Question Velocity & acceleration are both vectors. Is it possible for an object to have a zero velocity and a non-zero acceleration?

move or is turning around & changing Conceptual Question Velocity & acceleration are both vectors. Is it possible for an object to have a zero velocity and a non-zero acceleration? 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!

One-Dimensional Kinematics Examples

As already noted, the instantaneous acceleration is the average acceleration in the limit as the time interval becomes infinitesimally short. The instantaneous slope of the velocity versus time curve is the instantaneous acceleration. Figure 2-17. Caption: A graph of velocity v vs. time t. The average acceleration over a time interval Δt = t2 – t1 is the slope of the straight line P1P2: aav = Δv/ Δt. The instantaneous acceleration at time is t1 the slope of the v vs. t curve at that instant.

Example: Analyzing with graphs: The figure shows the velocity v(t) as a function of time for 2 cars, both accelerating from 0 to 100 km/h in a time 10.0 s. Compare (a) the average acceleration; (b) the instantaneous acceleration; & (c) the total distance traveled for the 2 cars. Solution: (a) Ave. acceleration: a = Both have the same ∆v & the same ∆t so a is the same for both. (b) Instantaneous acceleration: a = slope of tangent to v vs t curve. For about the first 4 s, curve A is steeper than curve B, so car A has greater a than car B for times t = 0 to t = 4 s. Curve B is steeper than curve A, so car B has greater a than car A for times t greater than about t = 4 s. Figure 2-19. Solution: a. Average acceleration is the same; both have the same change in speed over the same time. b. Car A accelerates faster than B at the beginning but then slower than B towards the end (look at the slope of the lines). c. Car A is always going faster than car B, so it will travel farther. (c) Total distance traveled: Except for t = 0 & t = 10 s, car A is moving faster than car B. So, car A will travel farther than car B in the same time.

Example: The figure shows the velocity v(t) as a function of time for 2 cars, both accelerating from 0 to 100 km/h in a time 10.0 s. Compare (a) the average acceleration; (b) the instantaneous acceleration; & (c) the total distance traveled for the 2 cars. Figure 2-19. Solution: a. Average acceleration is the same; both have the same change in speed over the same time. b. Car A accelerates faster than B at the beginning but then slower than B towards the end (look at the slope of the lines). c. Car A is always going faster than car B, so it will travel farther.

Example: The figure shows the velocity v(t) as a function of time for 2 cars, both accelerating from 0 to 100 km/h in a time 10.0 s. Compare (a) the average acceleration; (b) the instantaneous acceleration; & (c) the total distance traveled for the 2 cars. Figure 2-19. Solution: a. Average acceleration is the same; both have the same change in speed over the same time. b. Car A accelerates faster than B at the beginning but then slower than B towards the end (look at the slope of the lines). c. Car A is always going faster than car B, so it will travel farther. Solution: (a) Ave. acceleration: a = Both have the same ∆v & the same ∆t so a is the same for both.

Example: The figure shows the velocity v(t) as a function of time for 2 cars, both accelerating from 0 to 100 km/h in a time 10.0 s. Compare (a) the average acceleration; (b) the instantaneous acceleration; & (c) the total distance traveled for the 2 cars. Figure 2-19. Solution: a. Average acceleration is the same; both have the same change in speed over the same time. b. Car A accelerates faster than B at the beginning but then slower than B towards the end (look at the slope of the lines). c. Car A is always going faster than car B, so it will travel farther. Solution: (b) Instantaneous acceleration: a = slope of tangent to v vs t curve. For about the first 4 s, curve A is steeper than curve B, so car A has greater a than car B for times t = 0 to t = 4 s. Curve B is steeper than curve A, so car B has greater a than car A for times t greater than about t = 4 s. .

Example: The figure shows the velocity v(t) as a function of time for 2 cars, both accelerating from 0 to 100 km/h in a time 10.0 s. Compare (a) the average acceleration; (b) the instantaneous acceleration; & (c) the total distance traveled for the 2 cars. Figure 2-19. Solution: a. Average acceleration is the same; both have the same change in speed over the same time. b. Car A accelerates faster than B at the beginning but then slower than B towards the end (look at the slope of the lines). c. Car A is always going faster than car B, so it will travel farther. Solution: (c) Total distance traveled: Except for t = 0 & t = 10 s, car A is moving faster than car B. So, car A will travel farther than car B in the same time. .

Example (for you to work!) : Calculating Average Velocity & Speed Problem: Use the figure & table to find the displacement & the average velocity of the car between positions (A) & (F).

Example (for you to work!) : Graphical Relations between x, v, & a Problem: The position of an object moving along the x axis varies with time as in the figure. Graph the velocity versus time and acceleration versus time curves for the object.