Chapter 4: In this chapter we will learn about the kinematics (displacement, velocity, acceleration) of a particle in two or three dimensions. Projectile motion, Superposition principle Uniform Circular Motion Relative Motion 1-Mar-07 Dr. M. S. Kariapper

Displacement in a plane The displacement vector r: Displacement is the straight line between the final and initial position of the particle. That is the vector difference between the final and initial position. 1-Mar-07 Dr. M. S. Kariapper

Average Velocity Average velocity v: Average velocity: Displacement of a particle, Dr, divided by time interval Dt. 1-Mar-07 Dr. M. S. Kariapper

Instantaneous Velocity Instantaneous velocity : Limit of the average velocity as Dt approaches zero. The direction v is always tangent to the particles path. The instantaneous velocity equals the derivative of the position vector with respect to time. The magnitude of the instantaneous velocity vector is called the speed (scalar) 1-Mar-07 Dr. M. S. Kariapper

Checkpoint 2 The figure shows a circular path taken by a particle. If the instantaneous velocity of the particle is , through which quadrant is the particle moving when it is traveling (a) clockwise and (b) counterclockwise around the circle? 1-Mar-07 Dr. M. S. Kariapper

Average Acceleration Average acceleration: Average acceleration: Change in the velocity Dv divided by the time Dt during which the change occurred. Change can occur in direction and magnitude! Acceleration points along change in velocity Dv! 1-Mar-07 Dr. M. S. Kariapper

Instantaneous Acceleration Instantaneous acceleration: limiting value of the ratio as Dt goes to zero. Instantaneous acceleration equals the derivative of the velocity vector with respect to time. 1-Mar-07 Dr. M. S. Kariapper

Kinetic Quantities in 1-D and 2-D 1 Dimension 2 Dimension Displacement Average Velocity Inst. Velocity Average Acc. Inst. Acc. 1-Mar-07 Dr. M. S. Kariapper

Two- (or three)-dimensional motion with constant acceleration a Trick 1: The equations of motion we derived before (e.g. kinematic equations) are still valid, but are now in vector form. Trick 2 (Superposition principle): Vector equations can be broken down into their x- and y- components. Then calculated independently. Position vector: Velocity vector: 1-Mar-07 Dr. M. S. Kariapper

Two-dimensional motion with constant acceleration Velocity as function of time Position as function of time: 1-Mar-07 Dr. M. S. Kariapper

Sample Problem 4-5 What is ax and ay? A particle with velocity v=(-2.0i+4.0j)m/s at t=0 undergoes a constatnt acceleration a of magnitude a =3.0m/s2 at an angle q = 130° from the positive direction of the x-axis. What is the particle’s velocity v at t = 5.0s, in unit vector notation and as a magnitude and angle. What is ax and ay? 1-Mar-07 Dr. M. S. Kariapper

Projectile motion Two assumptions: Free-fall acceleration g is constant. Air resistance is negligible. The path of a projectile is a parabola (derivation: see book). Projectile leaves origin with an initial velocity of vo. Projectile is launched at an angle qo Velocity vector changes in magnitude and direction. Acceleration in y-direction (vertical) is -g. Acceleration in x-direction (horizontal) is 0. 1-Mar-07 Dr. M. S. Kariapper

Projectile motion Superposition of motion in x-direction and motion in y-direction Acceleration in x-direction is 0. Acceleration in y-direction is -g. (Constant velocity) (Constant acceleration) The horizontal motion and vertical motion are independent of each other; that is, neither motion affects the other. 1-Mar-07 Dr. M. S. Kariapper

Simultaneous fall demo Which ball will hit the ground first? Straight drop Straight out Both at the same time 1-Mar-07 Dr. M. S. Kariapper

A battleship simultaneously fires two shells at enemy ships. If the shells follow the parabolic trajectories shown, which ship gets hit first? A. B. C. Both hit at the same time. D. Need more information. 1-Mar-07 Dr. M. S. Kariapper

Hitting the bull’s eye. How’s that? Demo. Explanation using Simulation 1-Mar-07 Dr. M. S. Kariapper

Example for a Projectile Motion A stone was thrown upward from the top of a cliff at an angle of 37o to horizontal with initial speed of 65.0m/s. If the height of the cliff is 125.0m, how long is it before the stone hits the ground? Since negative time does not exist. 1-Mar-07 Dr. M. S. Kariapper

Example cont’d What is the speed of the stone just before it hits the ground? What are the maximum height and the maximum range of the stone? 1-Mar-07 Dr. M. S. Kariapper

Uniform Circular Motion  Motion in a circular path at constant speed. Velocity is changing, thus there is an acceleration!! Acceleration is perpendicular to velocity Centripetal acceleration is towards the center of the circle Magnitude of acceleration is r is radius of circle 1-Mar-07 Dr. M. S. Kariapper

Relative Motion O Frame A O’ Frame B P Results of Physical measurements in different reference frames could be different Consider that you are driving a car (reference frame B) with (constant) velocity relative to stationary frame A. To you (B), an object (P) in the car does not move while to the person (A) outside the car P is moving in the same speed and direction as your car is. O Frame A O’ Frame B P Since we consider only the case where is constant: 1-Mar-07 Dr. M. S. Kariapper

Moving frame of reference Relative Motion Moving frame of reference A boat heading due north crosses a river with a speed of 10.0 km/h. The water in the river has a speed of 5.0 km/h due east. In general we have Determine the velocity of the boat. If the river is 3.0 km wide how long does it take to cross it? 1-Mar-07 Dr. M. S. Kariapper

Performance Objective 1.  Express the position of an object  using vector notation. 2.  Express the velocity of an object  using vector notation. 3.  Express the acceleration of an object using vector notation. 4.  Analyze projectile motion using  x and y components. 5.  Explain what is meant by  uniform circular motion. 6.  Explain what is meant by  centripetal acceleration and force.. 7.  Discuss relative motion in one dimension. 8.  Discuss relative motion in two dimensions. 9.  Solve word problems involving two and three-dimensional motion. 1-Mar-07 Dr. M. S. Kariapper