Ying Yi PhD Chapter 2 Motion in One Dimension 1 PHYS HCC

Outline PHYS HCC 2 What is dynamics? Quantities in Motion  Displacement VS. Distance  Velocity VS. Speed  Acceleration 1D motion: Uniform acceleration  General case  Free fall

Position Defined in terms of a frame of reference A choice of coordinate axes Defines a starting point for measuring the motion One dimensional, so generally the x- or y-axis 3 PHYS HCC

Displacement Defined as the change in position f stands for final and i stands for initial Units are meters (m) in SI 4 PHYS HCC

5 Displacement Examples From A to B x i = 30 m x f = 52 m  x = 22 m The displacement is positive, indicating the motion was in the positive x direction From C to F x i = 38 m x f = -53 m  x = -91 m The displacement is negative, indicating the motion was in the negative x direction

Displacement, Graphical 6 PHYS HCC

Vector vs. Scalar 7 PHYS HCC Difference: Magnitude and direction Only Magnitude Notation: Examples: Displacement Mass, length

Displacement vs. Distance Displacement: Change in position Distance: actual path length Example: Throw a ball straight up and then catch it at the same point you released it The distance is? The displacement is? 8 PHYS HCC

A man is traveling along the equator from one side of earth to another side, what is the magnitude of displacement the man made? (Radius of earth is R.) PHYS HCC 9 A. 2*PI*R B. PI*R C. R D. 2*R

Velocity Definition: The average velocity is rate at which the displacement occurs Expression: SI Unit: m/s Note that: Velocity can be positive or negative;  t is always positive 10 PHYS HCC

Speed Definition: The average speed of an object is defined as the total distance traveled divided by the total time elapsed Expression: SI Unit: m/s Note that: Speed is a scalar quantity, only + value. 11 PHYS HCC

12 Velocity vs. Speed Q1: Which car have the greater average velocity? Q2: Which car has the greater speed?

Graphical Interpretation of Velocity Velocity can be determined from a position-time graph Average velocity equals the slope of the line joining the initial and final positions An object moving with a constant velocity will have a graph that is a straight line 13 PHYS HCC

14 Average Velocity, (Constant) The straight line indicates constant velocity The slope of the line is the value of the average velocity

Notes on Slopes The general equation for the slope of any line is The meaning of a specific slope will depend on the physical data being graphed Slope carries units 15 PHYS HCC

16 Average Velocity, (Non Constant) The motion is non- constant velocity The average velocity is the slope of the straight line joining the initial and final points

Instantaneous Velocity The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero The instantaneous velocity indicates what is happening at every point of time 17 PHYS HCC

Instantaneous Velocity on a Graph The slope of the line tangent to the position-vs.-time graph is defined to be the instantaneous velocity at that time The instantaneous speed is defined as the magnitude of the instantaneous velocity 18 PHYS HCC

Example 2.2 Velocity PHYS HCC 19 Andy Green in the car ThrustSSC set a world record of m/s in The car was powered by two jet engines, and it was the first one officially to exceed the speed of sound. To establish such a record, the driver makes two runs through the course, one in each direction, to nullify wind effects. Figure 2.3a shows that the car first travels from left to right and covers a distance of 1609 m in a time of s. Figure 2.3b shows that in the reverse direction, the car covers the same distance in s. From these data, determine the average velocity for each run.

Acceleration Changing velocity means an acceleration is present Acceleration is the rate of change of the velocity Units are m/s² (SI), cm/s² (cgs), and ft/s² (US Cust) 20 PHYS HCC

Average Acceleration Vector quantity When the sign of the velocity and the acceleration are the same (either positive or negative), then the speed is increasing When the sign of the velocity and the acceleration are in the opposite directions, the speed is decreasing 21 PHYS HCC

Negative Acceleration A negative acceleration does not necessarily mean the object is slowing down If the acceleration and velocity are both negative, the object is speeding up 22 PHYS HCC

Instantaneous and Uniform Acceleration The limit of the average acceleration as the time interval goes to zero When the instantaneous accelerations are always the same, the acceleration will be uniform The instantaneous accelerations will all be equal to the average acceleration 23 PHYS HCC

Graphical Interpretation of Acceleration Average acceleration is the slope of the line connecting the initial and final velocities on a velocity-time graph Instantaneous acceleration is the slope of the tangent to the curve of the velocity-time graph 24 PHYS HCC

Average Acceleration 25 PHYS HCC

26 Relationship Between Acceleration and Velocity Uniform velocity (shown by red arrows maintaining the same size) Acceleration equals zero

PHYS HCC 27 Relationship Between Velocity and Acceleration Velocity and acceleration are in the same direction Acceleration is uniform (blue arrows maintain the same length) Velocity is increasing (red arrows are getting longer) Positive velocity and positive acceleration

PHYS HCC 28 Relationship Between Velocity and Acceleration Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same length) Velocity is decreasing (red arrows are getting shorter) Velocity is positive and acceleration is negative

Motion Diagram Summary Please insert active figure PHYS HCC

Which of these graphs is not physically possible? PHYS HCC 30

Example 2.3 Acceleration PHYS HCC 31 Suppose the plane in Figure 2.4 starts from rest ( m/s) when t0=0 s. The plane accelerates down the runway and at t=29 s attains a velocity of km/h, where the plus sign indicates that the velocity points to the right. Determine the average acceleration of the plane.

Uniform acceleration motion 32 PHYS HCC

Graphical Interpretation of the Equation 33 PHYS HCC

Problem-Solving Hints Read the problem Draw a diagram Choose a coordinate system, label initial and final points, indicate a positive direction for velocities and accelerations Label all quantities, be sure all the units are consistent Choose the appropriate kinematic equation Solve for the unknowns You may have to solve two equations for two unknowns Check your results (Value and units) 34 PHYS HCC

Example 2.5 Uniform Acceleration PHYS HCC 35 The speedboat in Figure 2.8 has a constant acceleration of +2.0 m/s 2. If the initial velocity of the boat is +6.0 m/s, find the boat’s displacement after 8.0 seconds.

Group Problem PHYS HCC 36 The spacecraft shown in Figure 2.11a is traveling with a velocity of m/s. Suddenly the retrorockets are fired, and the spacecraft begins to slow down with an acceleration whose magnitude is 10.0 m/s 2. What is the velocity of the spacecraft when the displacement of the craft is +215 km, relative to the point where the retrorockets began firing?

Example2.9 PHYS HCC 37 A motorcycle ride consists of two segments, as shown in Figure 2.12a. During segment 1, the motorcycle starts from rest, has an acceleration of +2.6 m/s 2, and has a displacement of +120 m. Immediately after segment 1, the motorcycle enters segment 2 and begins slowing down with an acceleration of -1.5 m/s2 until its velocity is +12 m/s. What is the displacement of the motorcycle during segment 2?

Group Problem: Safe distance PHYS HCC 38 Dr. Yi is driving a Camry at a speed of 60.0mph on highway 59N. The car in front suddenly stopped. Dr. Yi immediately apply the brake, which creates an acceleration of -3.76m/s 2. Find out at least how far Dr. Yi should be away from the car ahead in order not to crash?

Free Fall (Constant Acceleration) PHYS HCC 39 All objects moving under the influence of gravity only are said to be in free fall All objects falling near the earth’s surface fall with a constant acceleration The acceleration is called the acceleration due to gravity, and indicated by g Galileo Galilei

Notes on Acceleration due to Gravity Symbolized by g g = 9.80 m/s² When estimating, use g ≈ 10 m/s 2 g is always directed downward Toward the center of the earth Ignoring air resistance and assuming g doesn’t vary with altitude over short vertical distances, free fall is constantly accelerated motion 40 PHYS HCC

41 Free Fall – an object dropped Initial velocity is zero Let up be positive Use the kinematic equations Generally use y instead of x since vertical Acceleration is g = m/s 2 v o = 0 a = g

PHYS HCC 42 Free Fall – an object thrown downward a = g = m/s 2 Initial velocity  0 With upward being positive, initial velocity will be negative

PHYS HCC 43 Free Fall -- object thrown upward Initial velocity is upward, so positive The instantaneous velocity at the maximum height is zero a = g = m/s 2 everywhere in the motion v = 0

Thrown upward, cont. The motion may be symmetrical Then t up = t down Then v = -v o The motion may not be symmetrical Break the motion into various parts Generally up and down 44 PHYS HCC

Example 2.10 Free Fall PHYS HCC 45 A stone is dropped from rest from the top of a tall building, as Figure 2.14 indicates. After 3.00s of free- fall, what is the displacement y of the stone? What is the velocity of the stone?

Group Problem: combined motion PHYS HCC 46 A rocket moves straight upward, starting from rest with an acceleration of m/s 2. It runs out of fuel at the end of 4.00s and continues to coast upward, reaching a maximum height before falling back to earth. (a) Find the rocket’s velocity and position at the end of 4.00s. (b) Find the maximum height the rocket reaches. (c) Find the velocity the instant before the rocket crashes on the ground.

Combination Motions 47 PHYS HCC

Chapter 2: Homework PHYS HCC 48 2,11,19,25,31,36,46,49,53,