Chapter 2: Motion in One Dimension Introduction  The study of motion and of physical concept such as force and mass is called dynamics.  The part of dynamics that describes motion without regard to its causes is called kinematics.  In this chapter, we will learn about kinematics in one dimension: motion along a straight line.

 Motion involves the displacement of an object from one place to another.  For that we need a convenient coordinate system and a specific origin : A frame of reference Displacement 0 x (m) displacement  x: final position initial position (origin)

 Displacement: Displacement 0 (origin) Point A to point B: Point C to point F: Displacement has both a magnitude and a direction: It’s a vector. A quantity that has both a magnitude and a direction is called a vector. A quantity that has only a magnitude and no direction is called a scalar.

 Velocity vs. speed Velocity Speed is a scalar quantity, while velocity is a vector quantity. Average speed of an object over a given time interval: Average velocity of an object over a given time interval: Average velocity vs. average speed SI unit: meter per second m/s average speed = v _ _

Velocity Point A to point B: Point C to point F:  Average velocity examples

Velocity Point A to point B:  Graphical interpretation of velocity slope Average velocity Instantaneous velocity derivative of x with respect to t at point A or slope of line tangent to point A

Velocity  Graphical interpretation of velocity

Acceleration  Average acceleration change of velocity change of time average accelerationSI unit: m/s 2

Acceleration  Average acceleration (cont’d) change of velocity change of time average acceleration =0.0 s =2.0 s =10 m/s =20 m/s

Acceleration  Instantaneous acceleration instantaneous acceleration The Instantaneous acceleration of an object at a given time equals the slope of the tangent to the velocity vs. time graph at that time.

 Examples Example 2.2: Slowly moving train a) Average velocity from O to C: b) Average velocity from O to A: d) Instantaneous velocity at t=2.00 s: d) Instantaneous velocity at t=9.00 s: 3.0 s 9.0 s 4.5 m Examples of Velocity and Acceleration

 Examples Example 2.3: Catching a fly ball a) Instantaneous acceleration at A: b) Instantaneous acceleration at B: c) Instantaneous acceleration at C:

One-dimensional Motion with Constant Acceleration  Instantaneous velocity When an object moves with constant acceleration, the instantaneous acceleration at any point in a time interval is equal to the value of the average acceleration over the entire time interval: Let

One-dimensional Motion with Constant Acceleration  Displacement When the velocity is increasing or decreasing uniformly with time, we can express the average velocity in any time interval as the arithmetic average of the initial velocity: Now from the definition of displacement: v

 Displacement The area under the graph of v vs. t for any object is equal to the displacement  x of the object. One-dimensional Motion with Constant Acceleration or

 Some examples Examples of Constant Acceleration Example 2.5: Car chase A trooper spots a speeding car at 24.0 m/s The trooper sets off in chase at a=3.00 m/s 2 a)How long does it take the trooper to overtake the speeding car? at time t m

Examples of Constant Acceleration A trooper spots a speeding car at 24.0 m/s The trooper sets off in chase at a=3.00 m/s 2 Solve: b) At t the trooper’s speed is: Example 2.5: Car chase (cont’d)

Examples of Constant Acceleration Example 2.7: Runaway length

 Effect of gravity on freely falling objects Freely Falling Objects When air resistance is negligible, all objects dropped under the influence of gravity near Earth’s surface fall toward Earth with the same constant acceleration. (Galileo Galilei, ~Year 1600) A freely falling object is any object moving freely under the influence of gravity alone, regardless of its initial motion. The magnitude of the free-fall acceleration is denoted by g. The value of g decreases with increasing altitude, and varies slightly with latitude, as well. At Earth’s surface, the value of g is approximately 9.80 m/s 2.

 Examples Freely Falling Objects Example 2.9: Not a bad throw for a rookie! y x a)Find the time when the stone reaches its maximum height (v=0). velocity: The velocity is zero at the maximum height:

Freely Falling Objects Example 2.9: Not a bad throw for a rookie! y x b) Determine the stone’s maximum height. y coordinate: The velocity is zero at the maximum height:

Freely Falling Objects Example 2.9: Not a bad throw for a rookie! y x c) Find the time the stone takes to return to its initial position (y=0) and its velocity. y coordinate : The y coordinate is zero at that moment : The velocity at that moment is :

Freely Falling Objects Example 2.9: Not a bad throw for a rookie! y x d) Find the time required for the stone to reach the ground (y=-50.0 m). y coordinate : The y coordinate is m at that moment :

Freely Falling Objects Example 2.10: A rocket goes ballistic y x a) Find the rocket’s velocity and position after 4.00 s. y coordinate at t=4.0 s : velocity at t=4.00 s : t=4.00 s

Freely Falling Objects Example 2.10: A rocket goes ballistic y x b) Find the maximum height the rocket attains. y coordinate : velocity is zero and a = g : t=4.00 s

Freely Falling Objects Example 2.10: A rocket goes ballistic y x c) Find the velocity of the rocket just prior to impact. y coordinate = 0.00 : velocity is zero and a = g : t=4.00 s