Physics 13 General Physics 1 Kinematics of Motion Motion along a Straight Line MARLON FLORES SACEDON

TOPICS: Motion along a Straight Line (Rectilinear) Introduction to Kinematics of Motion Xt-Graph, vt-graph, at-Graph Uniformly Accelerated Bodies (UAB) Free Falling Bodies (FFB) Motion in Two or Three Dimensions Velocity vectors and Acceleration vectors Projectile Motion Motion in a Circle

Motion along a Straight Line: Introduction What is Kinematics? A motion of an object without considering outside factors which causes their motion. Purely descriptive study of motion. What is motion? A change in position of particle at certain instance.

Motion along a Straight Line: Introduction How to describe motion? We can describe motion by… Position (𝑥) Time (𝑡) Displacement (∆𝑥) Velocity (𝑣) Acceleration (𝑎) x𝑡-graph, 𝑣𝑡-graph, & 𝑎𝑡-graph

Motion along a Straight Line: Introduction Describing motion using Time (t) and Position (x) x 1 , x 2 , x 3 … x n = position of particle from the ref point. −𝑥 4 − 𝑥 0 𝑥 1 − 𝑥 0 𝑥 𝑛 − 𝑥 0 𝑥 2 − 𝑥 0 𝑥 3 − 𝑥 0 𝑝 4 ( 𝑡 4 , −𝑥 4 ) 𝑝 1 ( 𝑡 1 , 𝑥 1 ) 𝑝 2 ( 𝑡 2 , 𝑥 2 ) 𝑝 3 ( 𝑡 3 , 𝑥 3 ) 𝑝 𝑛 ( 𝑡 𝑛 , 𝑥 𝑛 ) 𝑟𝑒𝑓 𝑡 4 𝑡 0 𝑡 1 𝑡 2 𝑡 3 𝑡 𝑛 − 𝑥 4 𝑥 0 𝑥 1 𝑥 2 𝑥 3 𝑥 𝑛 𝑥 2 − 𝑥 1 𝑥 3 − 𝑥 2 Displacement ∆𝑥 = change in position −𝑥 4 − 𝑥 3 𝑥 𝑛 −(− 𝑥 4 )

Motion along a Straight Line: 𝑥𝑡-Graph, 𝑣𝑡-Graph, & 𝑎𝑡-Graph Describing motion using Time (t) and Position (x) 𝑝 4 ( 𝑡 4 , −𝑥 4 ) 𝑝 1 ( 𝑡 1 , 𝑥 1 ) 𝑝 2 ( 𝑡 2 , 𝑥 2 ) 𝑝 3 ( 𝑡 3 , 𝑥 3 ) 𝑝 𝑛 ( 𝑡 𝑛 , 𝑥 𝑛 ) 𝑟𝑒𝑓 ( 𝑡 0 , 𝑥 0 ) Describing motion using 𝑥𝑡-Graph 𝑝 𝑛 ( 𝑡 𝑛 , 𝑥 𝑛 ) 𝑡 𝑛 𝑥 𝑛 𝑝 3 ( 𝑡 3 , 𝑥 3 ) 𝑡 3 𝑥 3 𝑝 2 ( 𝑡 2 , 𝑥 2 ) Position (𝑥) 𝑡 2 𝑥 2 𝑝 1 ( 𝑡 1 , 𝑥 1 ) 𝑥 1 𝑡 1 𝑡 4 - 𝑥 4 Time (𝑡) 𝑝 4 ( 𝑡 4 , 𝑥 4 )

Motion along a Straight Line: 𝑥𝑡-Graph, 𝑣𝑡-Graph, & 𝑎𝑡-Graph Describing motion with velocity 𝑥𝑡-Graph 𝑝 𝑛 ( 𝑡 𝑛 , 𝑥 𝑛 ) 𝑡 𝑛 𝑥 𝑛 Velocity of particle from pt.1 to pt.2 or 𝒗 𝟏𝟐 = ∆𝒙 ∆𝒕 𝑣 23 = 𝑥 3 − 𝑥 2 𝑡 3 − 𝑡 2 𝑝 3 ( 𝑡 3 , 𝑥 3 ) 𝑡 3 𝑥 3 𝑝 2 ( 𝑡 2 , 𝑥 2 ) Position (𝑥) ( 𝑥 3 − 𝑥 2 ) 𝑣 4𝑛 = 𝑥 𝑛 − 𝑥 4 𝑡 𝑛 − 𝑡 4 𝑡 2 𝑥 2 𝑣 12 = 𝑥 2 − 𝑥 1 𝑡 2 − 𝑡 1 ( 𝑡 3 − 𝑡 2 ) 𝑣 34 = 𝑥 4 − 𝑥 3 𝑡 4 − 𝑡 3 ( 𝑥 𝑛 − 𝑥 4 ) ( 𝑥 2 − 𝑥 1 ) 𝑝 1 ( 𝑡 1 , 𝑥 1 ) (𝑡 2 − 𝑡 1 ) 𝑥 1 𝑡 1 ( 𝑥 4 − 𝑥 3 ) 𝑡 4 - 𝑥 4 (𝑡 𝑛 − 𝑡 4 ) (𝑡 4 − 𝑡 3 ) Time (𝑡) 𝑝 4 ( 𝑡 4 , 𝑥 4 ) The ratio between change in position and time interval is average velocity. So, the slope between points is average velocity. The greater the slope, the faster the particle moving. If slope is positive, particle moves towards positive axis, and reverse direction for negative slope. If slope is zero, particle is at rest position.

Motion along a Straight Line: 𝑥𝑡-Graph, 𝑣𝑡-Graph, & 𝑎𝑡-Graph Describing motion with velocity 𝑥𝑡-Graph 𝑝 𝑛 ( 𝑡 𝑛 , 𝑥 𝑛 ) 𝑝 3 ( 𝑡 3 , 𝑥 3 ) 𝑣 23 = 𝑣 𝐵 𝑣 4𝑛 = 𝑣 𝐷 𝑝 2 ( 𝑡 2 , 𝑥 2 ) Position (𝑥) 𝑡 𝐵 = 𝑡 2 + 𝑡 3 2 𝑣 12 = 𝑣 𝐴 𝑣 34 = 𝑣 𝐶 𝑡 𝐷 = 𝑡 4 + 𝑡 𝑛 2 𝑝 1 ( 𝑡 1 , 𝑥 1 ) 𝑡 𝐴 = 𝑡 1 + 𝑡 2 2 𝑡 𝐶 = 𝑡 3 + 𝑡 4 2 Time (𝑡) 𝑝 4 ( 𝑡 4 , 𝑥 4 )

Motion along a Straight Line: 𝑥𝑡-Graph, 𝑣𝑡-Graph, & 𝑎𝑡-Graph Describing motion with acceleration 𝑥𝑡-Graph 𝑣 23 = 𝑣 𝐵 𝑣 4𝑛 = 𝑣 𝐷 Position (𝑥) 𝑡 𝐵 𝑣 12 = 𝑣 𝐴 𝑣 34 = 𝑣 𝐶 𝑡 𝐷 𝑡 𝐴 𝑡 𝐶 Time (𝑡) Time (𝑡) Velocity (𝑣) 𝑣𝑡-Graph 𝑣 𝐷 𝑡 𝐷 𝑣 𝐴 𝑡 𝐴 𝑣 𝐵 𝑡 𝐵 𝑣 𝐶 𝑡 𝐶

Motion along a Straight Line: 𝑥𝑡-Graph, 𝑣𝑡-Graph, & 𝑎𝑡-Graph Describing motion with acceleration Velocity of particle from pt.A to pt.B or 𝒂 𝑨𝑩 = ∆𝒗 ∆𝒕 𝑣𝑡-Graph Time (𝑡) Velocity (𝑣) 𝑣 𝐷 𝑡 𝐷 𝒂 𝑨𝑩 = 𝑣 𝐵 − 𝑣 𝐴 𝑡 𝐵 − 𝑡 𝐴 = ∆ 𝑣 𝐴𝐵 ∆𝒕 𝑣 𝐴 𝑡 𝐴 𝑣 𝐵 − 𝑣 𝐴 𝒂 𝑪𝑫 = 𝑣 𝐷 − 𝑣 𝐶 𝑡 𝐷 − 𝑡 𝐶 = ∆ 𝑣 𝐷𝐶 ∆𝒕 𝑣 𝐵 𝑡 𝐵 𝑡 𝐵 − 𝑡 𝐴 𝑣 𝐶 𝑡 𝐶 𝒂 𝑩𝑪 = 𝑣 𝐶 − 𝑣 𝐵 𝑡 𝑐 − 𝑡 𝐴 = ∆ 𝑣 𝐵𝐶 ∆𝒕 The ratio between change in velocity and time interval is average acceleration. So, the slope between velocities is average acceleration. The greater the slope, the greater the acceleration of the moving particle. If slope is positive, particle is accelerating, and decelerating for negative slope . If slope is zero, particle moves with constant velocity.

Motion along a Straight Line: 𝑥𝑡-Graph, 𝑣𝑡-Graph, & 𝑎𝑡-Graph Example: The following are points of rectilinear motion in a moving particle: 𝑝 1 (5,10), 𝑝 2 (10,45), 𝑝 3 (15, 90), 𝑝 4 (20,90), 𝑝 5 (25,−10). The units of time interval and positions are in seconds and meters respectively. Calculate the displacements and velocities between points. Then construct the 𝑥𝑡 & vt-Graph of the travel using this graph. Time 𝑡 𝑖𝑛 𝑠 Position 𝑥 𝑖𝑛 𝑚 10 20 30 40 50 60 70 80 90 −10 𝑝 3 𝑝 4 Time 𝑡 𝑖𝑛 𝑠 Velocity 𝑣 𝑖𝑛 𝑚/𝑠 10 15 5 20 25 10 15 5 20 25 𝑝 𝐵 1 −15 −10 −5 10 −20 𝑝 𝐴 𝑝 𝐶 𝑝 2 xt-Graph vt-Graph 𝑝 1 𝑝 𝐷 𝑝 5 𝑣 34 = 0 5 =0 𝑡 𝐴 = 5+10 2 =7.5𝑠 𝑡 𝐶 = 15+20 2 =17.5𝑠 𝑣 12 = 35 5 =7𝑚/𝑠 𝑥 12 =45−10=35𝑚 𝑡 12 =10−5=5𝑠 𝑥 23 =90−45=45𝑚 𝑡 23 =15−10=5𝑠 𝑣 23 = 45 5 =9𝑚/𝑠 𝑥 34 =90−90=0 𝑡 34 =20−15=5𝑠 𝑣 45 = −100 5 =−20𝑚/𝑠 𝑡 𝐵 = 10+15 2 =12.5𝑠 𝑡 𝐷 = 20+25 2 =22.5𝑠 𝑥 34 =−10−90=−100𝑚 𝑡 45 =25−20=5𝑠

Motion along a Straight Line: 𝑥𝑡-Graph, 𝑣𝑡-Graph, & 𝑎𝑡-Graph Assignment Describe your motion within 24 hours using Position & Time, displacements, average velocity, average acceleration, 𝑥𝑡−𝐺𝑟𝑎𝑝ℎ, 𝑣𝑡−𝐺𝑟𝑎𝑝ℎ, & a𝑡−𝐺𝑟𝑎𝑝ℎ. Your starting point will be here (our Class room) and your time starts NOW…. Note: Solutions should be written in Short sized bond paper. Use graphing paper for the Graphs.

To be continued…