Topic 2.1 Motion Kari Eloranta 2017 Jyväskylän Lyseon lukio August 18, 2017
Velocity and Speed 2.1: Kinematic Quanties: Displacement Definition of Displacement Displacement is the change in position. 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 2.1: Kinematic Quantities: Average Velocity Next we introduce the average velocity, which is not in the Data Booklet. 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 t 2 t 1 (2) 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 2.1: Kinematic Quantities: 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 2.1: Kinematic Quantities: Average Speed The average speed is not in the Data Booklet either. 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 2.1: Kinematic Quantities: 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 2.1: Kinematic Quantities: Instantaneous Velocity Instantaneous velocity is the velocity at a certain moment of time. The word velocity is a shorthand for 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 ). If we know the position of a moving object as a function of time, we can determine the instantaneous velocity from the 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 2.1: Kinematic Quanties: Instantaneous Speed Speed is the magnitude of velocity. Instantaneous speed is the speed at a certain moment of time. The word speed is a shorthand for 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 ).
Acceleration 2.1: Kinematic Quanties: Instantaneous Acceleration Instantaneous acceleration is the acceleration at a certain moment of time. The word acceleration is a shorthand for 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 ). If we know the velocity of a moving object as a function of time, we can determine the instantaneous acceleration from the graph by drawing a tangent line to the point of interest. The slope of the tangent is the instantaneous acceleration of the object at that point in time.
Acceleration Constant Acceleration 2.1: 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 t s Figure: The acceleration of a cart on an inclined plane for three inclinations. In each case, the velocity versus time graph is a straight line.
Acceleration Constant Acceleration 2.1: Uniformly Accelerated Motion When a cart moves down an incline, its velocity v is directly proportional to time t. 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 Exercise 1. Calculate the acceleration of the cart in A), B) and C). (4)
Acceleration Constant Acceleration 2.1: Position (Distance Fallen) in Free Fall m 5 s 4 3 2 1 t 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0s A falling basketball is in free fall at the beginning of the fall. The rate of change of position increases with time.
Acceleration Constant Acceleration 2.1: Velocity in Free Fall m/s v 10 9 8 7 6 5 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 t s Free fall is uniformly accelerated motion in which the rate of change of velocity with time is constant (acceleration is constant). In the graph, velocity v is (directly) proportional to time t, (v t).
Acceleration Constant Acceleration 2.1: 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. t
Acceleration Constant Acceleration 2.1: Area Under a Velocity vs Time Graph 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.
Kinematic Equations in Data Booklet (constant acceleration) 2.1: Kinematic Quanties: Average Velocity v av There are four equations in sub-topic 2.1 in the Data Booklet. The equations apply only to cases where the acceleration is constant (uniformly accelerated motion). We begin by defining the average velocity, which is not in the data booklet. Average Velocity The average velocity in uniformly accelerated motion is v av = u + v 2 where u is the initial velocity of the object, and v the final velocity. (5)
Kinematic Equations in Data Booklet (constant acceleration) 2.1: Kinematic Quantities: Displacement and Position s Displacement in Uniformly Accelerated Motion If an object travels time t at average velocity of v av = u+v, the displacement is 2 s = v av t = u + v 2 t (6) where u is the initial velocity of the object, and v the final velocity. Position in Uniformly Accelerated Motion If an object starts at the origin, and accelerates at constant acceleration a, after time t its position is where u is the initial velocity of the object. s = ut + 1 2 at 2 (7) Note! Initial velocity u, final velocity v and acceleration a may be positive or negative depending on the direction of motion.
Kinematic Equations in Data Booklet (constant acceleration) 2.1: Kinematic Quantities: Final Velocity and Speed in Uniformly Accelerated Motion Final Speed v in Uniformly Accelerated Motion If an object starts accelerating from the initial speed u, and accelerates the distance s with acceleration a, the square of the final speed v of the object is v 2 = u 2 + 2as. (8) Often, it is useful to know the relation between the final velocity v, initial velocity u and acceleration a in uniformly accelerated motion. Final Velocity v in Uniformly Accelerated Motion If an object starts accelerating from the initial speed u, and accelerates time t with acceleration a, the final velocity of the object is Note! The equation is derived from a = v u t. v = u + at. (9)