LEARNING OBJECTIVES By the end of this topic, you will be able to: Equations of motion ● define and use distance, displacement, speed, velocity and acceleration ● use graphical methods to represent distance, displacement, speed, velocity and acceleration ● determine displacement from the area under a velocity–time graph ● determine velocity using the gradient of a displacement–time graph ● determine acceleration using the gradient of a velocity–time graph ● derive, from the definitions of velocity and acceleration, equations that represent uniformly accelerated motion in a straight line ● solve problems using equations that represent uniformly accelerated motion in a straight line, including the motion of bodies falling in a uniform gravitational field without air resistance ● describe an experiment to determine the acceleration of free fall using a falling object ● describe and explain motion due to a uniform velocity in one direction and a uniform acceleration in a perpendicular direction How will you know if an object is in motion? Describing the position and motion of an object requires both a reference position from which reference directions are given and units that are understood by each person. These things are arbitrary. MOTION An object is in motion when it is continuously changing its position based on a reference point and as observed by a person or a device. A set of coordinates that can be used to determine positions (or change in position) of objects. POSITION, SPEED, VELOCITY, & ACCELERATION DISTANCE VS. DISPLACEMENT Distance Distance is defined as the total path length covered by an object that is in motion. DISTANCE VS. DISPLACEMENT Displacement DISTANCE VS. DISPLACEMENT If you go on a long walk, the distance you travel will be greater than your displacement. In this example, the walkers travel a distance of 15 km, but their displacement is only 10 km, because this is the distance from the start to the finish of their walk. Check your Understanding: Distance is equal to displacement. This statement is: (a) always true (b) sometimes true (c) never true SPEED VS. VELOCITY Check your Understanding: Are the values of speed and velocity always the same? If not, what is/are the situation/s where the magnitudes of speed and velocity are equal? Justify your answer. Average speed vs. Average velocity Instantaneous velocity Instantaneous velocity is the velocity of an object in motion at a specific point in time. Check your Understanding: What is your instantaneous speed at the instant you turn around to move in the opposite direction? (a) Depends on how quickly you turn around; (b) always zero; (c) always negative; (d) none of the above. Check your Understanding: Runner’s average velocity. The position of a runner as a function of time is plotted as moving along the x axis of a coordinate system. During a 3.00-s time interval, the runner’s position changes as shown in the figure. What is the runner’s average velocity? Check your Understanding: The figure shows the unusual path of a confused football player. After receiving a kickoff at his own goal, he runs downfield to within inches of a touchdown, then reverses direction and races back until he’s tackled at the exact location where he first caught the ball. DUring this run, which took 25 x, what is (a) the path length he travels, (b) his displacement, (c) his average velocity in the x-direction and (d) his average speed? Example: The Turtle and the Rabbit A turtle and a rabbit engage in a footrace over a distance of 4.00 km. The rabbit runs 0.500 km and then stops for a 90.0-min nap. Upon awakening, he remembers the race and runs twice as fast. Finishing the course in a total time of 1.75 h, the rabbit wins the race. (a) Calculate the average speed of the rabbit. (b) What was his average speed before he stopped for a nap? Assume no detours or doubling back. Acceleration The rate of change of velocity over time. ● ● ● Speeding up Slowing down Changing direction Check your Understanding: In which of the following cases does a car have negative velocity and a positive acceleration? A car that is travelling in _______. Acceleration Example A sports car accelerates along a straight test track from rest to 70 km h–1 in 6.3 s. What is its average acceleration? A railway train, travelling along a straight track, takes 1.5 minutes to come to rest from a speed of 115 km h–1. What is its average acceleration while braking? Acceleration Example A cyclist accelerates from 0 m/s to 28.8 km/h in 3 seconds. What is his acceleration? Is this acceleration higher than that of a car which accelerates from 0 to 108 km/h in 10 seconds? How long does it take for a car to change its velocity from 10.o m /s to 25.0 m /s if the acceleration is 5.00 m/s2? Acceleration Example Answer Questions 1,2 and 4-7 from pages 31 and 35 of your course book. Acceleration Example MOTION GRAPHS Three types of graph can represent motion: ● displacement-time graphs ● velocity-time graphs and ● acceleration-time graphs MOTION GRAPHS Activity Represent the motion of the following using graphs. 1. Increasing Speed in the Positive Direction 2. Decreasing speed in the positive direction 3. Increasing speed in the negative direction 4. Decreasing speed in the negative direction 5. Up and down the ramp 6. Up and down the ramp MOTION GRAPHS Displacement-time graphs ● Displacement-time graphs show the changing position of an object in motion ● They also show whether an object is moving forwards (positive displacement) or backwards (negative displacement) ○ A negative gradient = a negative velocity (the object is moving backwards) ● The gradient (slope) of a displacement-time graph is equal to velocity ○ The greater the slope, the greater the velocity MOTION GRAPHS Displacement-time graphs MOTION GRAPHS Displacement-time graphs ● On a displacement-time graph… ○ slope equals velocity ○ the y-intercept equals the initial displacement ○ a straight(diagonal) line represents a constant velocity ○ a curved line represents an acceleration ○ a positive slope represents motion in the positive direction ○ a negative slope represents motion in the negative direction ○ a zero slope (horizontal line) represents a state of rest ○ the area under the curve is meaningless MOTION GRAPHS Velocity-time graphs ● ● Velocity-time graphs show the speed and direction of an object in motion over a specific period of time The area under a velocity-time graph is equal to the displacement of a moving object displacement = area under a velocity-time graph MOTION GRAPHS Velocity-time graphs MOTION GRAPHS Velocity-time graphs ● On a velocity-time graph… ○ slope equals acceleration ○ the y-intercept equals the initial velocity ○ a straight line represents uniform acceleration ○ a curved line represents non-uniform acceleration ○ a positive slope represents positive acceleration ○ a negative slope represents negative acceleration ○ a zero slope (horizontal line) represents motion with constant velocity ○ the area under the curve equals the displacement or distance travelled MOTION GRAPHS Acceleration-time graphs ● On an acceleration-time graph… ○ slope is meaningless ○ the y-intercept equals the initial acceleration ○ a zero slope (horizontal line) represents an object undergoing constant acceleration ○ the area under the curve equals the change in velocity MOTION GRAPHS Example The velocity-time graph of a vehicle travelling with uniform acceleration is shown in the diagram below. Calculate the displacement of the vehicle at 40 s. MOTION GRAPHS Example The velocity-time graph of a vehicle travelling with uniform acceleration is shown in the diagram below. Calculate the displacement of the vehicle at 40 s. MOTION GRAPHS Example: The table below shows how the velocity of a motorcyclist changed during a speed trial along a straight road. a Draw a velocity–time graph for this motion. b From the table, deduce the motorcyclist’s acceleration during the first 10 s. c Check your answer by finding the gradient of the graph during the first 10 s. d Determine the motorcyclist’s acceleration during the last 15 s. e Use the graph to find the total distance travelled during the speed trial. MOTION GRAPHS MOTION GRAPHS The figure on the right shows the motion of a cart. a) Draw the position-time graph, velocity-time graph, and acceleration-time graph of the motion of the cart. b) Is the cart (i) speeding up, (ii) slowing down, or (iii) moving at constant speed? c) What is the average speed of the cart? d) What is the average velocity of the cart? e) What is the instantaneous velocity of the cart at t = 2 s? MOTION GRAPHS Example: A sailboat is sailing in a straight line with a velocity of 12 m/s. Then at time t = 0 s, a stiff wind blows causing the sailboat to accelerate as seen in the diagram. What is the velocity of the boat after the wind has blown for 9 s? KINEMATIC EQUATIONS OF MOTION ● The kinematic equations of motion are a set of four equations which can describe any object moving with constant acceleration FOUR KINEMATIC EQUATIONS Note: They can only be used: ■ for motion in a straight line ■ for an object with constant acceleration. KINEMATIC EQUATIONS OF MOTION ● It’s important to know where these equations come from and how they are derived: KINEMATIC EQUATIONS OF MOTION A graph showing how the velocity of an object varies with time KINEMATIC EQUATIONS OF MOTION KINEMATIC EQUATIONS OF MOTION The two terms ut and ½at2 make up the area under the graph KINEMATIC EQUATIONS OF MOTION KINEMATIC EQUATIONS OF MOTION FOUR KINEMATIC EQUATIONS KINEMATIC EQUATIONS OF MOTION Solving Problems with Kinematic Equations ● ● ● Step 1: Write out the variables that are given in the question, both known and unknown, and use the context of the question to deduce any quantities that aren’t explicitly given ○ e.g. for vertical motion a = ± 9.81 m s–2, an object which starts or finishes at rest will have u = 0 or v = 0 Step 2: Choose the equation which contains the quantities you have listed ○ e.g. the equation that links s, u, a and t is s = ut + ½at2 Step 3: Convert any units to SI units and then insert the quantities into the equation and rearrange algebraically to determine the answer KINEMATIC EQUATIONS OF MOTION Example The rocket shown in the figure on the right lifts off from rest with an acceleration of 20 m s−2. Calculate its velocity after 50 s. KINEMATIC EQUATIONS OF MOTION Example The car shown in the figure is travelling along a straight road at 8.0 m s−1. It accelerates at 1.0 m s−2 for a distance of 18 m. How fast is it then travelling? KINEMATIC EQUATIONS OF MOTION Example A race car starting from rest accelerates at a constant rate of 5.00 m/s2 . (a) What is the velocity of the car after it has traveled 30.5 m? (b) How much time has elapsed? KINEMATIC EQUATIONS OF MOTION Example Answer Questions 13-14 from page 40 KINEMATIC EQUATIONS OF MOTION Example FREE FALL ACCELERATION If you release an object, let’s say a baseball, in the air, what do you think will happen to the object? What caused it to fall? It’s GRAVITY! Because of the gravitational field of the Earth, the Earth exerts a force on all objects dropped near its surface. The gravitational field near the surface of the Earth is taken to be uniform, so all objects fall with the same uniform acceleration. Which of these two will hit the ground first if they are released from the same height and at the same time? FREE FALL ACCELERATION In the absence of air resistance, the motion of any falling object is found to be under a constant acceleration called the acceleration due to gravity g which has a value of -9.8 m/s2 FREE FALL ACCELERATION Free Fall motion is any motion of a body that undergoes constant acceleration equal to -9.8 m/s2 and where the force of gravity is the only force acting upon it. FREE FALL ACCELERATION MOTION ANALYSIS As the object rises: ● Velocity decreases ● Acceleration is constant -9.8 m/s2 ● Velocity at the maximum height is ZERO. FREE FALL ACCELERATION MOTION ANALYSIS As the object falls: ● Velocity increases ● Acceleration is constant -9.8 m/s2 FREE FALL ACCELERATION KINEMATICS EQUATION FREE FALL ACCELERATION EXAMPLE A rock is dropped 80 meters from a cliff. How long does it take to reach the ground? FREE FALL ACCELERATION EXAMPLE A cricketer throws a ball vertically upward into the air with an initial velocity of 18.0 m s–1. How high does the ball go? How long is it before it returns to the cricketer’s hands? FREE FALL ACCELERATION EXAMPLE A cricketer throws a ball vertically upward into the air with an initial velocity of 18.0 m s–1. How high does the ball go? How long is it before it returns to the cricketer’s hands? FREE FALL ACCELERATION EXAMPLE A ball is thrown vertically upward with a speed of 25.0 m s-1. (a) How high does it rise? (b) How long does it take to reach its highest point? (c) How long does the ball take to hit the ground after it reaches its highest point? (d) What is its velocity when it returns to the level from which it started? FREE FALL ACCELERATION EXAMPLE A 0.50 kg ball being held at rest above the ground is released. The ball begins to fall under only the effect of gravity. At the instant that the ball is 2.0 m above the ground, the speed of the ball is 2.5 m s-1. At what height was the ball released? FREE FALL ACCELERATION EXAMPLE An object is released from rest near a planet’s surface. A graph of the acceleration as a function of time for the object is shown for the 4 s after the object is released. The positive direction is considered to be upward. What is the displacement of the object after 2 s? FREE FALL ACCELERATION UNIFORMLY ACCELERATED MOTION A speedboat increases its speed uniformly from 25 m/s to 35 m/s in a distance of 250 m. Find (a) the magnitude of its acceleration and (b) the time it takes the boat to travel the 200-m distance. A hockey player is standing on his skates on a frozen pond when an opposing player, moving with a uniform speed of 15 m/s, skates by with the puck. After 4.0 s, the first player makes up his mind to chase his opponent. If he accelerates uniformly at 6.0 m/s 2 , (a) how long does it take him to catch his opponent, and (b) how far has he traveled in that time? Assume that the player with the puck remains in motion at constant speed. An apple drops from a tree and falls freely at a height of 3.5 m above the ground. Find the speed at which it reaches the ground. A bullet is shot vertically into the air with a velocity of 600 m/s. Neglecting air resistance, (a) How long is the bullet in the air? (b) How high does the bullet go?
