Chapter 3: Introduction to Kinematics Kari Eloranta 2018 Jyväskylän Lyseon lukio Pre Diploma Program Year October 11, 2017 1 / 17
3.1 Displacement Definition of Displacement Displacement is the change in position. Displacement The displacement is s = s 2 s 1 (1) where #» s 1 is the initial position and #» s 2 the final position of the object. Displacement is a vector quantity (it has magnitude and direction). The SI unit of displacement is one metre (1 m). In one-dimensional cases, displacement is either positive or negative, depending on the choice of coordinate system. Velocity and Speed 2 / 17
3.1 Velocity in Uniform Motion m s 4 3 2 1 0 1 1 2 3 4 5 6 7 8 The position s of a toy helicopter against time t. t s Velocity and Speed 3 / 17
3.2 Velocity in Uniform Motion Velocity Velocity is the rate of change of position with time. Consider the graph on the previous page. The position increases at constant rate in the time interval 0 s to 1.5 s. As a result, the velocity is a positive constant in that interval. The position does not change in the time interval 1.5 s to 5.0 s. As a result, the velocity is zero in that interval. The position decreases at constant rate in the time interval 5.0 s to 7.0 s. As a result, the velocity is a negative constant in that interval. Exercise 1. Calculate the velocity of the helicopter in each time interval. Velocity and Speed 4 / 17
3.3 Average Velocity Definition of Average Velocity Average velocity is the average rate of change of position with time. Average Velocity If an object is at an initial position s 1 at time t 1 and at a final position s 2 at time t 2, its average velocity in the time interval t 1 to t 2 is v av = s t = s 2 s 1 (2) t 2 t 1 where s is the displacement and t the time elapsed. Average velocity is a measure of how much per unit time the position on average changes in a time interval. Velocity and Speed 5 / 17
3.3 Average Velocity Average velocity is a vector quantity. The SI unit of average velocity is one metre per second (1 ms 1 ). If the displacement is zero, the average velocity is zero. For example, if a runner completes a 400 m circular track in 58 s, her average velocity is zero. In one-dimensional motion, average velocity is either positive or negative depending on the direction of motion. Velocity and Speed 6 / 17
3.3 Average Speed Definition of Average Speed Average speed is the average rate of change of distance with time. Average Speed If an object travels a distance s in time t, its average speed is v = s t. (3) Average speed is a measure of how long a distance per unit time the object travels on average in a time interval. Velocity and Speed 7 / 17
3.3 Average Speed Average speed is a scalar quantity. The SI unit of average speed is one metre per second (1 ms 1 ). If a runner completes a 400 m circular track in 58 s, her average speed is v = s t = 400m 58s 6.9ms 1. Velocity and Speed 8 / 17
3.3 Instantaneous Velocity Instantaneous velocity is the velocity at a certain moment of time. The word velocity usually means instantaneous velocity. Definition of Velocity Velocity is the rate of change of position with time. Velocity is a vector quantity. The SI unit of velocity is one metre per second (1 ms 1 ). We can determine the instantaneous velocity from a position against time graph by drawing a tangent line to the point of interest. The slope of the tangent is the instantaneous velocity of the object at that point in time. In one-dimensional motion, velocity is either positive or negative depending on the direction of motion. Velocity and Speed 9 / 17
3.3 Instantaneous Speed Speed is the magnitude of velocity. Instantaneous speed is the speed at a certain moment of time. The word speed usually means instantaneous speed. Definition of Speed Speed is the rate of change of distance with time. Speed is a scalar quantity. The SI unit of speed is one metre per second (1 ms 1 ). Velocity and Speed 10 / 17
3.4 Instantaneous Acceleration Instantaneous acceleration is the acceleration at a certain moment of time. The word acceleration usually means instantaneous acceleration. Definition of Acceleration Acceleration is the rate of change of velocity with time. Acceleration is a vector quantity. The SI unit of acceleration is one metre per second squared (1 ms 2 ). The slope of the tangent line on a velocity against time graph is the instantaneous acceleration of the object at that point in time. If an object travels in positive direction, positive acceleration indicates increasing speed and negative acceleration indicates decreasing speed. Accelerated Motion 11 / 17
3.4 Uniformly Accelerated Motion ms 1 v 1.0 0.8 A) B) C) 0.6 0.4 0.2 0 0 0.5 1.0 1.5 Figure: The velocity of a cart on an inclined plane for three inclinations. In each case, the velocity versus time graph is a straight line. t s Accelerated Motion Uniform Acceleration 12 / 17
3.4 Uniformly Accelerated Motion When a cart rolls down an incline from rest, its velocity v is directly proportional to time t (v t or v = at). This is an example of uniformly accelerated motion. Uniformly Accelerated Motion In uniformly accelerated motion the acceleration is constant. Acceleration in Uniformly Accelerated Motion When an object accelerates uniformly from initial velocity u to final velocity v in time t, its acceleration is a = v u t (4) Exercise 2. Calculate the acceleration of the cart in A), B) and C). Explain your reasoning. Accelerated Motion Uniform Acceleration 13 / 17
3.4 Position (Distance Fallen) in Free Fall m 5 s 4 3 2 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 s t A falling ball is in free fall at the beginning of the fall. The rate of change of position increases with time. Accelerated Motion Uniform Acceleration 14 / 17
3.4 Velocity and Acceleration in Free Fall ms 1 10 9 8 7 6 5 4 3 2 1 0 v 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Free fall is uniformly accelerated motion in which the rate of change of velocity with time is constant (acceleration a is constant). The slope of the line equals the acceleration of free fall (m = a = g = v t ), where g = 9.81ms 2 is the acceleration due to gravity. Accelerated Motion Uniform Acceleration 15 / 17 t s
3.4 Area Under a Velocity vs Time Graph v The graph represents the velocity v of an object as a function of time t. The area of each rectangle is A i = v i t i, which has the unit of distance. Because in uniform motion the distance travelled is s = vt, the area of a rectangle approximates the actual distance travelled by the object in time t i. Accelerated Motion Area Under Velocity Graph 16 / 17 t
3.4 Area Under Velocity against Time Graph (cont.) v t Calculating the total area of the rectangles approximates the area under the graph. Increasing the number of rectangles increases the accuracy of the process. Accelerated Motion Area Under Velocity Graph 17 / 17