Sabtu, 19 November 2011

Motion in two dimensions

MOTION IN TWO DIMENSIONS

In this chapter we explore the kinematics of particles moving in two dimensions. We begin by studying in more detail the nature of the position vector, velocity and acceleration.

DISPLACEMENT, VELOCITY AND ACCELERATION VECTORS

Displacement
Displacement is a vector and the particle displacement is the difference between final and initial position. We now define the displacement vector for the particle Figure 4.1 as the difference between the final position vector and the initial position vector.
Direction Δr is shown in Figure 4.1. As we see from the picture, the amount less than the distance traveled along the curved path followed by the particles.

VELOCITY VECTOR
average speed of particles during a time interval Δt as the particle displacement divided by time interval

Instantaneous velocity v is defined as the limit of an average speed of Δr / Δt as close to zero:

ACCELERATION VECTOR
Knowing the speed at points allows us to determine the average acceleration of the particle as it moves is defined as the change in instantaneous velocity vector v divided by the interval of time during which the changes occur
When the average acceleration of particles changes during different time intervals, it is useful to determine the instantaneous acceleration. Instantaneous acceleration is defined as the limiting value of the ratio close to zero

In other words, instantaneous acceleration equal to the derivative of the velocity vector with respect to time.

TWO DIMENSIONAL MOTION WITH CONSTANT VELOCITY

Two-dimensional motion during acceleration remained constant in both magnitude and direction. It would be useful to analyze some common types of motion.
The vector position of a particle moving in the xy plane:

r = x + y
Where x, y and r change with time as a particle moving
i and j remain constant

velocity vector as a function of time
If the position vector is known, the particle velocity can be obtained from equations 4.3 and 4.6, which gives

Hence, substituting from equation 2.9 into the equation 4.7

Graphical representation Equation 4.8

Vf is generally not along the direction of either Vi or because this is the relationship between the amount of expression vector. We can write in component form

The position vector as a function of time X and y coordinates of a particle moving with constant acceleration

Substituting this expression into the equation r = 4.6 x i + y j which gives

This equation tells us that the position vector is the vector sum of the original position displacement arising from the initial particle velocity and displacement resulting from particle acceleration constant

Graphical representation Equation 4.9

Rf generally not along the direction of Vi or expression vector. We can write them in component form

PROJECTILE MOTION

Anyone who has watched a baseball moving projectile motion has been observed. The ball moves in a curved path, and motion are simple to analyze if we make two assumptions: (1) free-fall acceleration g is constant over the range of motion and directed downward and
(2) the effect of air resistance is negligible.dengan this assumption, we find that the path of projectiles, which we call the track, always a parabola.
Therefore, x and y components of the initial velocity is: