Kinematics

Velocity and Acceleration

Motion

• Change in position of object in relation to things that are considered stationary

• Usually earth is considered stationary

• Nothing is truly stationary (earth travels 108,000 km/hr orbiting sun)

• All motion is relative: must be related to other objects called your frame of reference

Distance and Displacement• Distance: how far object moves without

respect to direction, a scalar quantity• Displacement: change of position in a

particular direction; How far and in what direction object is from original position, a vector

• Both use symbol d ,and often x or y for 1-dim motion

• The unit is the meter

Speed

• The time rate of motion, the rate of change of position--a scalar

• Units of distance /time: m/s usually but can be miles/hr, km/hr; Symbol is v

• Average speed = total distance/elapsed time

• Instantaneous speed: rate of change of position at any instant

Velocity• Speed in a particular direction, a vector; Unit same

as speed; Symbol v

• Must include a direction, using angle from known reference points, compass headings, or just left & right, + & -,up & down

• Can be negative (going backwards)

• Average velocity = total displacement / elapsed time

Velocity• Instantaneous velocity: instantaneous

speed with current direction

• Constant velocity means no change of speed or direction

• Often we are interested in only the speed (we may know the direction) so speed and velocity are sometimes used interchangeably

Acceleration• Time rate of change of velocity; A

vector; Symbol a and units of m/s/s usually shortened to m/s2

• Acceleration can be negative

• Average acceleration = change in velocity / elapsed time for the change

• Galileo first to understand acceleration

1st Constant Accel. Equation

• If acceleration is constant, instantaneous acceleration always equals avg acceleration

• Use definitions of avg velocity and accel to calculate final velocity or distance

• Since a = (vf - vi)/t , then vf = vi + at

• If vi = 0 , then vf = at

• Use when distance not given or asked for

2nd Constant Acceleration Equation

• vavg = (vf + vi)/2 ; but also vavg = d/t ; so (vf + vi)/2 = d/t

• Now using our first equation for vf we can get (vi + vi + at)/2 = d/t

• Solving for d: d = vit + 1/2 at2

• If vi = 0, d = 1/2 at2

• Use when final speed not given or asked for

3rd Constant Acceleration Equation

• Solve 1st equation for t and substitute into 2nd equation, expand squared quantity and combine terms.

• Get 2ad = vf 2- vi2; solve for vf 2

• vf 2= vi2 + 2ad

• If vi = 0, vf 2= 2ad

• Use when time is not given or asked for

Graphing Motion: d vs t• Plot time as independent variable

• On position vs time graph, slope at any value of t gives instantaneous velocity

• If graph is linear, slope and v are constant

• If graph is curved, slope and v are found by drawing tangent line to curve and finding its slope

Graphing Motion: d vs t

• Uniform motion (constant velocity)

Graphing Motion: d vs t (x vs t)

• Accelerated motion (increasing velocity)

Graphing Motion: v vs t

• Slope of v vs t graph gives acceleration

• If graph is linear, acceleration is constant

• If graph is curved, instantaneous acceleration is found using slope of tangent line at any point

Tangent LineA line that just touches a curve at one point and gives the slope of the curve at that point.

Velocity vs Time: acceleration

Comparing Uniform and Accelerated Motion Graphs

Uniform motion Accelerated Motion

Comparing Uniform and Accelerated Motion Graphs

Uniform motion Accelerated Motion

Comparing Positive and Negative Velocity

Speeding up and Slowing Down

Velocity vs Time Graphs: Finding Displacement

• Displacement can be found from velocity graph by finding the area between the line of the graph and the time axis

• Divide the area bounded by the graph line, the horizontal axis and the initial and final times into geometric sections (squares, rectangles, triangles, trapezoids) and find the area

• Area below the time axis is negative displacement

Area under (enclosed by) the Velocity Graph

Area Enclosed by the Velocity Graph

• Divide complex areas into triangles and rectangles

Area Enclosed by the Acceleration Graph

• If acceleration vs. time is plotted, area between the graph line and the horizontal (time) axis gives the change in velocity that took place during the time interval

Free Fall• Common situation for constant acceleration is

free fall• Force of gravity causes falling bodies to

accelerate• Force varies slightly from place to place but

average acceleration is 9.80 m/s2 designated by symbol g

• Often for simplicity or approximations, g = 10 m/s2 is used

Free Fall• Distance increases with each

second of falling.• Object will fall 4.9 m (about 5

m) during the 1st second• Distance increases by 9.8m

(about 10 m) each second• Speed increases by 9.8 m/s

(about 10 m/s) for each second of falling

Keeping Track of the Signs• If motion is only in one direction (usually

down), using positive and negative signs to indicate direction is not necessary.

• With up and down motion, up is considered positive and down negative

• g must be negative (-9.80 m/s2) in these situations along with downward displacements and velocities

Air Resistance and Free Fall

• If air drag is ignored, all objects fall at the same rate

• Air resistance slows rate of fall, depending on object’s surface area, shape, texture and density of air

• For our purposes, air resistance is negligible

Equations for Free Fall

• Can use all constant acceleration equations for free fall

• Equations for vertical motion are written with symbol g in place of a and y in place of x or d

• Be careful with positive and negative signs!

Constant Acceleration Equations• Horizontal Motion • Vertical Motion

tavv if 2

21 )( tatvx i

xavv if 222

tgvv if 2

21 )( tgtvy i

ygvv if 222

12 i fx v v t 1

2 i fy v v t