Chapter 2 Motion in one dimension Kinematics Dynamics

Chapter 2 Motion in one dimension Kinematics Dynamics

Section 2 -3 Position, velocity and acceleration vectors 1. Position vector z At any particular time t, the particle can be located by its x, y and z coordinates, which are three components of the position vector : O where , and are the cartesian unit vectors. x Fig 2 -11 y

2. Displacement (位移) y We defined the displacement vector as the change in position vector from t 1 to t 2. t= t= O x z Fig 2 -12 Note: 1) Displacement is not the same as the distance traveled by the particle. 2) The displacement is determined only by the starting and ending points of the interval.

Then the displacement is Direction: from start point to end point

The relationship between : In general, Can ? Yes, for two cases: 1) 1 D motion without changing direction 2) When after take limit: and

The difference between : magnitude of : the change of length of position vectors Note and ( ):

When after take limit:

3. velocity and speed a. The average velocity in any interval is defined to be displacement divided by the time interval, (2 -7) when we use the term velocity, we mean the instantaneous velocity. b. To find the instantaneous velocity, we reduce the size of the time interval and then. , that is (2 -9)

In cartesian coordinates: The vector can also be written in terms of its components as: (2 -11) (2 -12)

Discussion The position vector of a moving particle at a moment is. The magnitude of the velocity of the particle at the moment is: （A） √ （B） （C） √ （D）

c. The terms average speed (平均速率) and speed（速率）: Average speed: is the total distance traveled. Thus, Speed:

d. Acceleration We define the average acceleration as the change in velocity per unit time, or (2 -14) And instantaneous acceleration (2 -16) By analogy with Eq (2 -12) , we can write the components acceleration vector as (2 -17)

Sample problem 2 -4 A particle moves in the x-y plane and , where , , and. Find the position, velocity, and acceleration of the particle when t=3 s.

Solution:

Sample problem How do the velocity and the acceleration vary with time if position x(t) is known? x(t)

Colonel J. P. Stapp was in his braking rocket sled Can you tell the direction of the acceleration from the figures? His body is an accelerometer not a speedometer. Out in

Section 2 -4 One-dimensional kinematics In one-dimensional kinematics, a particle can move only along a straight line. We can describe the motion of a particle in two ways: with mathematical equations and with graphs.

1. Motion at constant velocity Suppose a puck (冰球) moves along a straight line, which we will use as the x-axis. x A 0 t (a) B 0 (b) Fig 2 -15 t

2. Accelerated motion (变速运动) Two examples of accelerated motion are (2 -20) （2 -21）

2 -5 Motion with constant acceleration Let’s assume our motion is along the x axis, and represents the x component of the acceleration. and x(t) from Can we obtain or Note: the initial velocity in the calculation. ? (2 -26) must be known

It is the similar way to find x(t) from v(t). or Note: the initial position x 0 and velocity must be known in the calculation.

Discussion Relationship between , and Derivative Integral ( )

2 -6 Freely falling bodies …… Aristotle (384 -322 B. C. ) thought the heavier objects would fall more rapidly because of their weight. Galileo (1564 -1642) made correct assertion, that in the absence of air resistance all objects fall with the same speed. In 1971, astronaut David Scott dropped a feather and a hammer on the airless Moon, and he observed that they reached the surface at about the same time.

‘Feather and hammer’ experiment carried on the airless moon ‘Feather and apple’ exp. conducted in vacuum lab.

Niagara Fall Is it dangerous to free fall from its top? ! If the height of the fall is 48 m, how long will the person reach the fall bottom and what is his final speed?