Basic References: Tipler, P. and Mosca, G.(2004) Physics for Scientists and Engineers. Vth Ed. Freeman and Company Beer, F. and Jonhston, E. R. (2007) Vector Mechanics for Engineers. McGraw-Hill MECHANICS Kinematics of Particles: Kinematics is the study of the geometry of motion; is used to relate displacement, velocity, acceleration and time, without reference to the cause of motion. Motion in One Dimension Motion in Two and Three Dimensions

Displacement, Velocity, and Speed MECHANICS Kinematics of Particles Motion in One Dimension

Position, Displacement, Velocity and Speed Motion in One Dimension Position x i. is defined by a frame of reference Displacement: The change in the position of particle, m Velocity: The rate at which the position change, m/s Traveled distance: s total distance traveled Average speed: ratio of the total distance traveled to the total time from start to finish Instantaneous speed

Average and Instantaneous velocity. Motion in One Dimension The magnitude of the instantaneous velocity is the instantaneous speed

Acceleration MECHANICS Kinematics of Particles Motion in One Dimension

Acceleration Motion in One Dimension Acceleration is the rate of the change of instantaneous velocity Average acceleration, m/s 2 Instantaneous acceleration, m/s 2 Motion under constant acceleration, a = a av = constant ; uniformly accelerated rectilinear motion Practical Case: Objects in Free- fall “falling freely under the influence of gravity only” “Near the earth´s surface all unsupported objects fall vertically with constant acceleration (provided air resistance is negligible) Gravity acceleration ~ 9,81 m/s 2. Eliminating t

How is the motion that is showed in the figures? Write the equation that describes the position and acceleration in each case. Describe the motion as is represented in the figure. Estimate the acceleration at t=50 s; t=120 s; What is the traveled distance since the origin of time to t = 60 s; t= 180 s A car accelerates from rest as showed in the figure. Draw a graph displaying its velocity and position versus time Exercises

Relative Motion of two particles Motion in One Dimension Relative position coordinate of B with respect to A Relative velocity of B with respect to A Relative acceleration of B with respect to A Dependent motions The motions of particles are linked, they are not independent Examples: Pulleys and objects linked by inextensible strings

A student throws her cap straight upward with an initial speed of 14.7 m/s. (a) How long does it take it highest point; (b) What is the distance to the highest point ? (c) Assuming the cup is caught at the same height from which it was released, what is the total time the cap is in flight? Problems

A ball is thrown vertically upward from the 12-m level in an elevator shaft with an initial velocity of 18 m/s. At the same instant an open- platform elevator passes the 5-m level, moving upward with a constant velocity of 2 m/s. Determine (a) when and where the ball will hit the elevator, (b) the relative velocity of the ball with respect to the elevator when the ball Problems

MECHANICS Kinematics of Particles Motion in Two and Three Dimensions

Vectors Addition and substraction Parallelogram method

Vector rectangular components Vectors Components

Addition of vectors can be done by the head-to-tail, the parallelogram method, or analytically, using vector components Vectors

Summary of properties of vectors

3-3 Position, Velocity, and Acceleration

Position Vector, Velocity and acceleration Curvilinear Motion

A student throws her cap into the air with an initial velocity of 24,5 m/s at 36.9º above of horizontal. The path described by the cap is showed in the figure. Draw the position vector, velocity and acceleration at time t= 0, t=1, t= 2 and t = 3 (in seconds). Consider the motion of a pendulum bob shown in figure. Draw the position vector, the velocity and the acceleration at different times, t0, t2, t4, t6 and t8. A particle is moving in a circle with constant speed. Draw a diagram motion about it displaying the position vector, velocity and acceleration in different positions of the path. Exercises Curvilinear Motion

Rectangular components of position vector, velocity and acceleration Curvilinear Motion