INTRODUCTION. 1. One-Dimensional Kinematics

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1 INTRODUCTION Mechanics is the area of physics most apparent to us in our everyday lives Raising an arm, standing up, sitting down, throwing a ball, opening a door etc all governed by laws of mechanics Mechanics is the study of how objects move, how they respond to external forces, and how other factors such as mass and size affect their motion Derived from the Greek word µηχανική Topics covered: Position, Distance and Displacement Average Speed and Velocity Instantaneous Velocity Acceleration Motion with Constant Acceleration Applications of the Equations of Motion Freely Falling Objects 1

2 POSITION, DISTANCE AND DISPLACEMENT (1) We will treat all physical objects as point particles Meaning that we will consider all the mass of an object to be concentrated at a single point If you are interested in calculating the time it takes the Earth to complete a revolution around the Sun, it is reasonable to consider the Earth and Sun as simple particles (points) First step in describing the motion of a particle is to set up a coordinate system In one dimension, this is simply an x-axis with an origin (x = 0) and an arrow indicating the positive direction which is the direction in which x increases Once coordinate system is chosen, there must be consistency in the calculations Below, the particle is the person moving from left to right from initial position x i to final position x f Positive direction is to the right, so x f > x i 2

3 POSITION, DISTANCE AND DISPLACEMENT (2) The chosen coordinate system can be used to investigate a situation as shown above You leave your house, drive to a local store, then return home Distance covered is 8.6 miles, where distance is defined as the total length of travel with SI units metre, m (1 mile = 1609m) Distance is always positive because it has no direction associated with it (it is a scalar) Another useful way to characterise a particle s motion is in terms of the displacement, x change in position x = x f x i (metres, m) x is positive if final position is to the right of initial x is negative if final position is to the left of initial It is zero if x f = x i x is a vector, which has displacement and direction denoted by its sign Study figure above for examples of distance and displacement 3

4 AVERAGE SPEED AND EXAMPLES We need to consider how rapidly an object moves The simplest way to characterise the rate of motion is with the average speed = distance/elapsed time Dimensions are distance per time (m/s, kph, mph) Distance and elapsed time are positive, thus average speed is always positive Example: A kingfisher dives from a height of 7.0m with an average speed of 4.0m/s. How long does it take for it to reach the water? Example: You drive 4.0 miles at 30mph and then another 4.0 miles at 50mph. Is your average speed for the 8 mile trip greater than, equal to or less than 40mph? 4

5 AVERAGE VELOCITY AND EXAMPLES Average velocity, v av, - displacement per time Average velocity = displacement/elapsed time Algebraically: v av = x/ t = (x f x i )/(t f t i ) (m/s) The average velocity tells us how fast something is moving, and also the direction the object is moving If the object moves in the positive direction, x f > x i, so v av is positive If the object moves in the negative direction, x i > x f, so v av is negative Average velocity is a one-dimensional vector and its direction is given by its sign It gives more information than average speed and thus is used more frequently in physics Example: An athlete sprints 50.0m in 8.0s, stops, and then walks slowly back to the starting line in 40.0s. If the sprint direction is taken to be positive, what is (a) the average sprint velocity, (b) the average walking velocity and (c) the average velocity for the complete round trip 5

6 GRAPHICAL INTERPRETATION OF AVERAGE VELOCITY (1) It is often useful to visualise a particle s motion by sketching its position as a function of time Above shows a particle moving back and forth along x axis The position of the particle is indicated at a variety of times A more practical graph is that of position, x, (vertical axis) versus time, t (horizontal axis) An x versus t graph makes it easier to visualise a particle s motion 6

7 GRAPHICAL INTERPRETATION OF AVERAGE VELOCITY (2) Suppose you want to know the average velocity of the particle from t = 0 to t = 3s v av = x/ t = (2m - 1m)/(3s 0) = +0.33m/s To relate this to the x v t plot, draw a straight line connecting position at t = 0 and t = 3 (a) Slope of this line is x/ t (average velocity) The slope of a line that connects two points on an x v t plot is equal to v av during that time interval Between t = 2s and t = 3s there is a negative slope so v av = -2m/s (b) 7

8 INSTANTANEOUS VELOCITY (1) Computation for average velocity can miss a lot of information For example covering 200km in 2 hours in a car means your average velocity is 100kph, yet your speed throughout this journey is not constant (i.e. stopping, driving at maximum speed limit, etc) To get a more accurate representation, the average velocity should be taken over shorter periods of time Calculating the average velocity every 5 mins gives a more accurate picture than calculating it every 10 mins More accurate picture given when calculating the average velocity every minute or every second Ideally it is best to know the velocity of the particle at each instant in time Instantaneous velocity is the velocity corresponding to an instant in time 8

9 INSTANTANEOUS VELOCITY (2) Mathematically: SI units m/s v = lim x t 0 t lim t 0 means that the average velocity, x/ t, is evaluated over shorter and shorter time intervals, approaching zero in the limit Instantaneous velocity can be either positive or negative, or zero Instantaneous speed is the magnitude of the instantaneous velocity, as given by a speedometer As t becomes smaller, so does x and the ratio x/ t approaches a constant value For straight line plots of x v t, the velocity is constant, i.e. the average velocity is constant over any time interval and is equal to the instantaneous velocity at any time 9

10 INSTANTANEOUS VELOCITY (3) Generally a particle s velocity varies with time, and thus the x v t plot is not a straight line (above) What is the instantaneous velocity at t = 1.0s? First approximation: take average velocity between t = 0s and t = 2.0s (x i = 0, x f = 27.4m), v av = 13.7m/s The other green lines under the curve give v av over smaller time intervals (0.25s-1.75s; 0.5s-1.5s; 0.75s- 1.25s corresponding to 12.1m/s, 10.9m/s, 10.2m/s respectively) The smaller the time interval, v av tends to 10.0m/s, and so we conclude that the instantaneous velocity at t = 1.0s is 10.0 m/s The outer green line is the tangent line at t = 1.0s, and its slope is 10 m/s 10

11 INSTANTANEOUS VELOCITY (4) From the previous slide, it is clear that the average velocity lines have slopes that approach the slope of the tangent line as the time intervals become smaller The instantaneous velocity at a given time is equal to the slope of the tangent line at that point on an x versus t plot The average velocity is the slope of the straight line connecting two points corresponding to a given time interval Instantaneous velocity is the slope of the tangent line at a given instant of time 11

12 AVERAGE ACCELERATION Acceleration is the rate of change of velocity with time An object accelerates whenever its velocity changes (velocity can increase or decrease) The study of acceleration is central to most of physics Galileo showed that falling objects move with constant acceleration Newton showed the acceleration and force are linearly related Average acceleration a av = v/ t = (v f - v i )/(t f - t i ) m/s 2 The acceleration of gravity on the Earth s surface is 9.81m/s 2 which means that the velocity of falling objects changes by 9.81m/s every second! Average acceleration can be either positive or negative (deceleration), and is also a 1-dimensional vector Example: Saab advertises a car that goes from 0 to 60mph in 6.2s. What is the average acceleration of this car? An aeroplane has an average acceleration of 5.6m/s 2 during takeoff. How long does is take for it to reach 150mph? 12

13 INSTANTANEOUS ACCELERATION Just as we considered the limit of smaller and smaller time intervals to find an instantaneous velocity, we can do the same to define instantaneous acceleration Definition: a v = lim t 0 t Instantaneous acceleration is also a one-dimensional vector and its direction is given by its sign If the acceleration is constant, it has the same value at all times When acceleration is constant, the instantaneous and average accelerations are the same 13

14 GRAPHICAL INTERPRETATION OF ACCELERATION Acceleration can be measured graphically using a velocity versus time graph (v v t) The average acceleration of a v v t plot is the slope of a straight line connecting points corresponding to two different times The instantaneous acceleration is the slope of the tangent line at a particular time Example: A train moving in a straight line with an initial velocity of 0.5m/s accelerates at 2.0m/s 2 for 2.0s, coasts with zero acceleration for 3.0s and then accelerates at -1.5m/s 2 for 1.0s. What is the final velocity of the train? What is the average acceleration of the train? 14

15 VELOCITY AND ACCELERATION DIRECTIONS In one-dimension, nonzero velocities and accelerations are either positive or negative, depending on whether they point in the positive or negative direction of the coordinate system chosen Therefore the velocity and acceleration of an object may have the same or opposite signs When the velocity and acceleration of an object have the same sign, the speed of the object increases. In this case, the velocity and acceleration point in the same direction When the velocity and acceleration of an object have opposite signs, the speed of the object decreases. In this case, the velocity and acceleration point in opposite directions 15

16 VELOCITY AND ACCELERATION DIRECTIONS: EXAMPLE A ferry makes a short run between two ports. As the ferry approaches Guemes Island (travelling in the positive x direction, its speed is 7.4m/s. If the ferry slows to a stop in 12.3s, what is its average acceleration? As the ferry returns to Anacortes, its speed is 7.3m/s. If it comes to rest in 13.1s, what is its average acceleration? 16

17 MOTION WITH CONSTANT ACCELERATION In the next few slides, we will derive equations describing the motion of particles moving with constant acceleration These equations of motion can be used to describe a wide range of everyday phenomena In an ideal world with no resistance (such as air resistance, friction), moving bodies have constant acceleration Recall that if a particle has constant acceleration, the same acceleration at every instant in time, then its instantaneous acceleration, a, is equal to its average acceleration, a av a av = (v f v i )/(t f t i ) = a If t i = 0, and v i = v 0 at time zero, and simplifying the notion so that v f = v and t f = t a av = (v v 0 )/(t 0) = a Rearranging gives v v 0 = a(t 0) = at So the equation of constant acceleration of motion where velocity is a function of time is v = v 0 + at (straight line on v v t plot) The line crosses velocity axis at v 0, has slope a Example: A ball is thrown straight upward with an initial velocity of +8.2m/s. If the acceleration of the ball is m/s 2, what is the velocity after 0.5s and 1.0s? 17

18 AVAERAGE VELOCITY & POSITION AS FUNCTION OF TIME How far does a particle move in a given time if its acceleration is constant? Recall that v av = x/ t = (x f x i )/(t f t i ) Letting x i = x 0 and x f = x, t i = 0, t f = t Gives v av = (x x 0 )/(t 0) Thus x x 0 = v av (t 0) = v av t Or x = x 0 + v av t Above applies whether acceleration is constant or not, and a more useful expression is obtained by writing v av in terms of final and initial velocities as shown above From graph, v av = ½(v 0 + v) average velocity Thus x = x 0 + ½(v 0 + v)t Position as function of time Above equations only applies with constant acceleration (see right figure above) 18

19 AVAERAGE VELOCITY & POSITION AS FUNCTION OF TIME: EXAMPLE A boat moves slowly inside a marina (so as not to leave a wake) with a constant speed of 1.5m/s. As soon as it passes the breakwater, leaving the marina, it throttles up and accelerates at 2.4m/s 2. How fast is the boat moving after accelerating for 5.0s? How far has the boat travelled in this time? 19

20 POSITION AS A FUNCTION OF TIME IN TERMS OF CONSTANT ACCELERATION The velocity of the boat from the previous slide is plotted as a function of time As can be seen, the distance travelled by the boat is equal to the corresponding area under the velocity versus time curve General result: The distance travelled by an object from time t 1 to time t 2 is equal to the area under the velocity curve between those two times Recall that v = v 0 + at and v av = ½(v 0 + v) So ½(v 0 + v) = ½[v 0 + (v 0 + at)] = v 0 + ½ at Thus x = x 0 + ½(v 0 + v)t = x 0 + (v 0 + ½ at)t So the expression for position versus time in terms of constant acceleration a is x = x 0 + v 0 t + ½ at 2 20

21 POSITION AS A FUNCTION OF TIME: EXAMPLE A drag racer starts from rest and accelerates at 7.4m/s 2. How far has it travelled in 1.0s, 2.0s and 3.0s? The graph above plots x v t for the drag racer, and is parabolic due to the ½ at 2 term When a > 0, plot points upwards, and when a < 0, it curves downwards The greater the magnitude of a, the greater the curvature If a particle moves with constant velocity, a = 0, the t 2 dependence vanishes, and the x v t plot becomes a straight line 21

22 EQUATION OF MOTION RELATING VELOCITY TO POSITION Recall that v = v 0 + at, so t = (v v 0 )/a Recall that x = x 0 + ½(v 0 + v)t Substituting for t gives x = x 0 + ½(v 0 + v)[(v v 0 )/a)] (v 0 + v)(v v 0 ) = v 0 v v 02 + v 2 v 0 v = v 2 v 0 2 This x = x 0 + [(v 2 v 02 )/2a Thus v 2 = v a(x x 0 ) = v a x The above equation allows is to relate the velocity at one position to the velocity at another position, without knowing how much time is involved Example: Airliners at Heathrow accelerate from rest at one end of a runway, and must attain take off speed before reaching the other end of the runway. Plane A has acceleration a and take off speed v to. What is the minimum length of runway, x A, required for this plane? Plane B has the same acceleration as plane A, but needs twice the takeoff speed. Find x B and compare with x A. Find the minimum runway length for plane A if a = 2.2m/s 2 and v to = 95.0m/s (values typical for a 747). 22

23 SUMMARY OF CONSTANT ACCELERATION EQUATIONS OF MOTION 23

24 APPLICATIONS OF THE EQUATIONS OF MOTION (1) A park ranger driving down a country lane suddenly sees a deer frozen in the headlights. The ranger, who is driving at 11.4m/s, immediately applies the brakes and slows with an acceleration of 3.8m/s 2. If the deer is 20m from the vehicle when the brakes are applied, how close does it come to hitting the dear? How much time is needed for the vehicle to stop? 24

25 APPLICATIONS OF THE EQUATIONS OF MOTION (2) Above is a plot of the velocity of the ranger s vehicle, v, versus its position, x The vehicle comes to rest with constant acceleration, which means that its velocity decreases uniformly with time However, the velocity does not decrease uniformly with distance If the final velocity is 0 (i.e. when the vehicle comes to a rest), then x = -v 02 /2a 25

26 APPLICATIONS OF THE EQUATIONS OF MOTION (3) A joy rider doing 40mph (17.9m/s) in a 25mph zone approaches a parked police car. The instant the joy rider passes the police car, the police begin their pursuit. If the joy rider maintains a constant velocity, and the police car accelerates with a constant acceleration of 4.5m/s 2, how long does it take the police car to catch up? How far have the two cars travelled in this time? What is the velocity of the police car when it catches the joy rider? 26

27 FREELY FALLING OBJECTS The most famous example of motion with constant acceleration is free fall i.e. under the influence of gravity Objects of different weight fall with the same acceleration, provided air resistance is small enough to be ignored If a feather and a coin are both dropped in a vacuum, they will fall at the same rate Dropping a sheet of paper to the ground and a rubber ball, the paper will take longer because it has a larger surface area that is in contact with the air (air resistance) If the paper is scrunched up, it will fall to the ground in almost the same time as the rubber ball Free fall is the motion of an object subject only to the influence of gravity An object is in free fall as soon as it is released, whether it is dropped from rest, thrown downward or thrown upward Acceleration due to earth s gravity is g and varies according to the location on Earth, but for simplicity we will use 9.81m/s 2 27

28 FREELY FALLING OBJECTS: EXAMPLE (1) A person steps off the end of a 3.0m high diving board and drops to the water below. How long does it take for the person to reach the water? What is the person s speed on entering the water? 28

29 FREELY FALLING OBJECTS: EXAMPLE (2) A volcano shoots out blobs of molten lava from its crater. A geologist observing the eruption uses a stopwatch to time the flight of a particular lava bomb that is projected straight upward. If the time for it to rise and fall back to its launch height is 4.75s, and its acceleration is 9.81m/s 2 downward, what is its initial speed? 29

30 FREELY FALLING OBJECTS: EXAMPLE (3) A hot air balloon is rising straight upward with a constant speed of 6.5m/s. when the basket of the balloon is 20.0m above the ground, a bag of sand tied to the basket comes loose. How long is the bag of sand in the air before it hits the ground? What is the greatest height of the bag of sand during its fall to the ground? Note that the bag is moving upward with the balloon at the time it comes loose. 30