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

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 Oscillations = Moving (vibrating) back & forth on same path. –Chapter 11 THE COURSE THEME IS NEWTON’S LAWS OF MOTION!! Continuous Media Waves, Sound –Chapters 11,12 Fluids = Liquids & Gases –Chapter 10 Conservation Laws Energy, Momentum, Angular Momentum Newton’s Laws expressed in other forms!

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

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.

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

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, 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.

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) 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 1 = 10 m, x 2 = 30 m Displacement  ∆x = x 2 - x 1 = 20 m ∆  Greek letter “delta” meaning “change in” The arrow represents the displacement (meters). t1t1 t2t2  times

x 1 = 30 m, x 2 = 10 m Displacement  ∆x = x 2 - x 1 = - 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,...

The Text uses BOLD letters to denote vectors. I usually denote vectors 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.

Average Velocity Average Speed  (Distance traveled)/(Time taken) A Scalar A Vector 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  (Displacement)/(Time taken)

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

 times t2t2 t1t1 Bar denotes average ∆x = x 2 - x 1 = displacement ∆t = t 2 - t 1 = elapsed time Average Velocity (x 2 - x 1 )/(t 2 - t 1 ) General Case

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 v ave = (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  The velocity at any instant of time  The Average Velocity over an infinitesimally short time. Mathematically, the Instantaneous Velocity is formally defined as: Instantaneous Velocity  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 at any instant of time. Mathematically, instantaneous velocity is: Mathematicians call this a derivative.  Instantaneous Velocity v ≡ Time Derivative of Displacement x

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

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  = (v 2 - v 1 )/(t 2 - t 1 ) –a – Acceleration is a vector. Instantaneous acceleration Units: velocity/time = distance/(time) 2 = m/s 2

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 a = A

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 a = = (25 m/s – 0 m/s)/5 s = 5 m/s 2

Conceptual Question Velocity & Acceleration are both vectors. 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 the driver hits the brakes) is v 1 = 15.0 m/s. It takes t = 5.0 s to slow down to v 2 = 5.0 m/s. Calculate the car’s average acceleration. a = = (v 2 – v 1 )/(t 2 – t 1 ) = (5 m/s – 15 m/s)/(5s – 0s) a = m/s 2

Deceleration “Deceleration”: 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! 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.

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

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. Is it possible for an object to have a zero velocity and a non-zero acceleration? 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! Conceptual Question

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.

Solution: (a) Ave. acceleration: a = Both have the same ∆v & the same ∆t so a is the same for both. (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: 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. ( 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 (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.