Motion in Two Dimensions

1. Motion in TwoDimensions In this chapter, we explore the kinematics of a particle moving in two dimensions. We begin by studying in greater detail the vector natureof position, velocity, and acceleration. We then treat projectile motion and uniform circularmotion as special cases of motion in two dimensions.

2. The Position, Velocity, and Acceleration Vectors Displacement vector Average velocity Instantaneous velocity The magnitude of the instantaneous velocity vector of a particle is calledthe speed of the particle, which is a scalar quantity. The average acceleration of a particle is defined as the change in its instantaneous velocity vector Ddivided by the time interval Dtduring which that change occurs: Average acceleration

3. Instantaneous acceleration A particle moves from position A toposition B. Its velocity vector changes from to . The vector diagrams at the upper right show twoways of determining the vector Dfrom the initial and final velocities. Two-Dimensional Motion with Constant Acceleration Any Example ??? The motion in two dimensions can be modeled as two independent motions in each of the twoperpendiculardirections associated with the x and y axes. That is, any influence in the y directiondoes not affect the motion in the x direction and vice versa.

4. The position vector for a particle moving in the xyplane can be written where x, y, and change with time as the particle moves while the unit vectors remain constant. the velocity of the particle Velocity vector as a function of time for a particleunder constant acceleration in two dimensions Similarly, Position vector as a function of time for a particle under constant acceleration in two dimensions

5. We can consider these equations to be the mathematical representation of a two-dimensional version of the particle under constant acceleration model. Graphical representations Figure Vector representations and components of (a) thevelocity and (b) the position of a particle under constant acceleration in two dimensions.

6. Example S o l u t i o n ??

7. B) S o l u t i o n C) S o l u t i o n Use

8. Projectile Motion Anyone who has observed a baseball in motion has observed projectile motion.The ball moves in a curved path and returns to the ground. Projectile motion ofan object is simple to analyze if we make two assumptions: (1) the free-fall acceleration is constant over the range of motion and is directed downward (2) the effect of air resistance is negligible the path of a projectile, which we call its trajectory, isalways a parabola = where the initial x and y components of the velocity of the projectile are

10. Horizontal Range and Maximum Height of a Projectile We can determine h

11. We can determine h

12. Example Solutions ??

13. A) B)

14. Example S o l u t i o n ??

15. B) S o l u t i o n Use this component with the horizontal component to find the speed of the stone at

16. Example

17. Analysis Model: Particle in Uniform Circular Motion Figure 4.15a shows a car moving in a circular path; we describe this motion by callingit circular motion. If the car is moving on this path with constant speed v, wecall it uniform circular motion. Because it occurs so often, this type of motion isrecognized as an analysis model called the particle in uniform circular motion. The uniform circular motion, the accelerationvector can only have a component perpendicular to the path, which is towardthe center of the circle. The figure also shows the vector representing the change in position Dfor an arbitrary time interval.

18. An acceleration of this nature is called a centripetal acceleration (centripetal meanscenter-seeking). The subscript on the acceleration symbol reminds us that the acceleration is centripetal. Centripetal acceleration for a particle in uniform circular motion Period of circular motion for a particle in uniform circular motion We can express the centripetal acceleration of a particle in uniform circularmotion in terms of angular speed by combining this eq. angular speed

19. Particle in Uniform Circular Motion

20. Tangential and Radial Acceleration At any point on a circle, you can pick two special directions: The direction that points directly away from the center of the circle (along the radius) is called radial direction, and the direction that is perpendicular to this is called the tangential direction.  Just because an object moves in a circle, it has a centripetal acceleration ac, directed toward the center. We know this centripetal acceleration is given by ac = v2 / r This centripetal acceleration is directed along a radius so it may also be called the radial acceleration ar The tangential acceleration is, indeed, tangent to the path of the particle's motion. If the speed is not constant, then there is also a tangentialacceleration at.

21. Radial acceleration Total acceleration Tangential acceleration

23. Example

24. Relative Motion, Speed, Velocity and Acceleration An object is in motion when it is continuously changing its position relative to a reference point and as observed by a person or detection device. For example, you can see that an automobile is moving with respect to the ground. Motion is relative All motion is relative to the observer or to some fixed object. For example, when you see a bus drive by, it is moving with respect to you. However, if you are in a car that is moving in the same direction, the bus will be moving at a different velocity with respect to you. If your car is moving in the same direction and same speed as the bus, the bus will appear to not move with respect to you. Of course, if you compare the speed with the ground, both of you will be moving at some velocity.

25. Point of reference In talking about motion, it is important to indicate your point of reference. Supposea car was traveling at 60 miles per hour (mph) and hit another car, but there was hardly a dent. The reason could be that the second car was traveling in the same direction at 59 mph, so the car was going only 1 mph with respect to the second car when it hit it. observer B measures A to be moving to theleft with a velocity = with respect to time,noting that is constant, we obtain ; Galilean velocity transformation

26. Example

27. Summary