Kinematics of Particles Lecture II

Subjects Covered in Kinematics of Particles Rectilinear motion Curvilinear motion Rectangular coords n-t coords Polar coords Relative motion Constrained motion

Introduction Kinematics: Branch of dynamics that describes the motion of bodies without reference to the forces which either cause the motion or are generated as a result of the motion. Applications: The design of cams, gears, linkages, and other machines elements to control or produce certain desired motion, and the calculation of flight trajectories for aircraft, rockets, etc. Particle: As mentioned, a particle is a body whose physical dimensions are so small compared with the radius of curvature of its path, e.g. an aircraft and its flight path. Studying the Motion: Studying the motion of a body includes studying its displacement from one location to another, its velocity, and its acceleration.

Introduction (Cont.) Choice of Coordinates: The position of a particle P at any time t can be described by specifying its rectangular coordinates ( x, y, z ), its cylindrical coordinates ( r, , z ), or its spherical coordinates ( R, ,  ). The motion of P can also be described by measurements along the tangent ( t ) and normal ( n ) to a curved path. These two are path variables since they move with the particle on the path. The motion of a body described by fixed reference axes known as absolute motion, while the motion described by a moving reference axes known as relative motion.

Rectilinear Motion Studying the motion of a particle moving in a straight line (1-D) -S+S Displacement

Rectilinear Motion - Velocity Average velocity ( v av ): Instantaneous velocity ( v ): as  t approaches zero in the limit, which is or Note: the velocity is positive or negative depending on the displacement (1)

Rectilinear Motion – Acceleration Average acceleration ( a av ): Instantaneous acceleration ( a ): as  t approaches zero in the limit, which is or Note: the acceleration is positive or negative depending whether the velocity is increasing or decreasing or (2) -S+S v1v2  v

Rectilinear Motion – Acceleration (Cont.) Velocity & acceleration: are vector quantities, as we will see in the study of curvilinear motion; however, since in rectilinear motion, the particle is moving in straight line path, the sense of direction is described by plus or minus sign. To obtain differential equation relating displacement, velocity, and acceleration: dt has to be eliminated from equation (2) (using Chain rule) Equations (1), (2), and (3): known as the differential equations for the rectilinear motion. or (3)

Rectilinear Motion – Graphical Interpretation The net displacement of a particle during interval  t : The net change in velocity of a particle during interval  t : When the acceleration is a function of the position coordinates S : or

Rectilinear Motion – Problems Classifications Rectilinear Motion (Problems Classifications) Given s(t) Required v(t) and/or a(t) Given a a (t) Required v(t) and/or s(t) a (v) Required v(t) or v(s) and/or s(t) a (s) Required v(s) and/or s(t) a = constant Given v v (t) Required s(t) and/or a(t) v = constant or

Exercises

Exercise # 1 The position of a particle which moves along a straight line is defined by the relation, s = t 3 - 6t t + 40, where s is expressed in meters and t in seconds. Determine (a) the time at which the velocity will be zero, (b) the position and distance traveled by the particle at that time, (c) the acceleration of the particle at that time, (d) the distance traveled by the particle from t = 4 s to t = 6 s.

Exercise # 2 A ball is tossed with a velocity of 10 m/s directed vertically upward from a window located 20 m above the ground. Knowing that the acceleration of the ball is constant and equal to 9.81 m/s 2 downward, determine (a) the velocity v and elevation y of the ball above the ground at any time t, (b) the highest elevation reached by the ball and the corresponding value of t, (c) the time when the ball will hit the ground and the corresponding velocity. Draw the v-t and y-t curves.

Exercise # 3 A rocket travel upward at 75m/s. When it is 40m from the ground, the engine fails. Determine max height s B reached by the rocket and its speed just before it hits the ground.

A bicycle moves along a straight road such that its velocity is described by the graph as shown. What is the total distance travelled by the bicycle. Construct the a-t graph for 0 ≤ t ≤ 30s. Exercise # 4