Chapter 2 Motion Motion

Motion Want to describe bodies in motion and how they behave in time Want to describe bodies in motion and how they behave in time Speed vs velocity Speed vs velocity Speed is a scalarSpeed is a scalar Velocity is a vectorVelocity is a vector examplesexamples

Speed The average speed of an object is defined as the total distance traveled divided by the total time elapsed The average speed of an object is defined as the total distance traveled divided by the total time elapsed Speed is a scalar quantitySpeed is a scalar quantity SI units are m/sSI units are m/s

Velocity It takes time for an object to undergo a displacement It takes time for an object to undergo a displacement The average velocity is rate at which the displacement occurs The average velocity is rate at which the displacement occurs SI units are m/s SI units are m/s

Example Car travels 120 km 45° N of E in 3 hours.

Conversion Factors Multiply by “1” until you get the right units Multiply by “1” until you get the right units Length: in, ft, m, mile, km, light year,Length: in, ft, m, mile, km, light year, Time: s, min, h, day,Time: s, min, h, day,

More on speed and velocity Motion by graphs Motion by graphs Slope Slope Motion with constant speed (velocity) Motion with constant speed (velocity) Motion with changing velocity Motion with changing velocity Average velocity Average velocity Instantaneous velocity Instantaneous velocity Instantaneous speed is the Mag. Of instantaneous velocity Instantaneous speed is the Mag. Of instantaneous velocity

Acceleration Changing velocity (non-uniform) means an acceleration is present Changing velocity (non-uniform) means an acceleration is present Acceleration is the rate of change of the velocity Acceleration is the rate of change of the velocity Units are m/s² (SI), cm/s² (cgs), and ft/s² (US Cust) Units are m/s² (SI), cm/s² (cgs), and ft/s² (US Cust)

Acceleration Vector quantity Vector quantity Can be constant Can be constant Will mainly study uniform acceleration Will mainly study uniform acceleration Negative acceleration means the object is slowing down (when v is positive) Negative acceleration means the object is slowing down (when v is positive) Graphs Graphs

Example Car accelerates uniformly from 0 to 60 miles/hour (East) in 6 s.

Uniformly Acceleration Motion Along a line (one dimension) Along a line (one dimension) Simplest case Simplest case Free fall Free fall A lot of examples A lot of examples Extend to projectile motion (2 dim) Extend to projectile motion (2 dim)

Linear Motion Summary (1) (1) (2) (2) (3) (3) (4) (4) (5) (5)

Example Car moving at 25 m/s hits the brakes & stops in a distance of 25 m 1.What was the car’s (uniform) deceleration? 2.How long did it take to stop?

Example A car is moving at a constant speed of 90 km/h when it begins to pass a train moving in the opposite direction at 60 km/h. If the train is 200 m long, How long does it take the car to pass the train?How long does it take the car to pass the train? How far did the car travel?How far did the car travel? How far did the train travel?How far did the train travel?

Free Fall All objects moving under the influence of gravity only are said to be in free fall All objects moving under the influence of gravity only are said to be in free fall Free fall does not depend on the object’s original motionFree fall does not depend on the object’s original motion All objects falling near the earth’s surface fall with a constant acceleration All objects falling near the earth’s surface fall with a constant acceleration The acceleration is called the acceleration due to gravity, and indicated by g The acceleration is called the acceleration due to gravity, and indicated by g

Acceleration due to Gravity Symbolized by g Symbolized by g g = 9.80 m/s² g = 9.80 m/s² When estimating, use g 10 m/s 2When estimating, use g 10 m/s 2 acc is always directed downward acc is always directed downward toward the center of the earthtoward the center of the earth Ignoring air resistance and assuming g doesn’t vary with altitude over short vertical distances, free fall is constantly accelerated motion Ignoring air resistance and assuming g doesn’t vary with altitude over short vertical distances, free fall is constantly accelerated motion

Free Fall – an object dropped Initial velocity is zero Initial velocity is zero Let up be positive Let up be positive Use the equations Use the equations Generally use y instead of x since verticalGenerally use y instead of x since vertical Acceleration is g = 9.80 m/s 2 Acceleration is g = 9.80 m/s 2 v o = 0 a = - g

Free Fall – an object thrown downward a = m/s 2 a = m/s 2 Initial velocity  0 Initial velocity  0 With upward being positive, initial velocity will be negativeWith upward being positive, initial velocity will be negative

Free Fall -- object thrown upward Initial velocity is upward, so positive Initial velocity is upward, so positive The instantaneous velocity at the maximum height is zero The instantaneous velocity at the maximum height is zero a = m/s 2 everywhere in the motion a = m/s 2 everywhere in the motion v = 0

Thrown upward, cont. The motion may be symmetrical The motion may be symmetrical Then t up = t downThen t up = t down Then v = -v IThen v = -v I The motion may not be symmetrical The motion may not be symmetrical Break the motion into various partsBreak the motion into various parts Generally up and down Generally up and down

Free Fall (g=9.8 m/s²) (1) (1) (2) (2) (3) (3) (4) (4) (5) (5)

Example A ball is launched straight upward with an initial speed of 20 m/s. How long does it take to return?How long does it take to return? How high did it go?How high did it go?

Projectile Motion Example of motion in 2-dim Example of motion in 2-dim An object may move in both the x and y directions simultaneously An object may move in both the x and y directions simultaneously It moves in two dimensionsIt moves in two dimensions The form of two dimensional motion we will deal with is called projectile motion The form of two dimensional motion we will deal with is called projectile motion v I and  0 v I and  0

Projectile Motion (2-dim) Horizontal motion is independent of vertical motion Horizontal motion is independent of vertical motion Break motion into 2 separate parts Break motion into 2 separate parts Horizontal: x componentHorizontal: x component Vertical: y componentVertical: y component Vertical motion: free fall Vertical motion: free fall Horizontal motion: constant velocity Horizontal motion: constant velocity

Assumptions of Projectile Motion We may ignore air friction We may ignore air friction We may ignore the rotation of the earth We may ignore the rotation of the earth With these assumptions, an object in projectile motion will follow a parabolic path With these assumptions, an object in projectile motion will follow a parabolic path

Rules of Projectile Motion The horizontal motion (x) and vertical of motion (y) are completely independent of each other The horizontal motion (x) and vertical of motion (y) are completely independent of each other The x-direction is uniform motion The x-direction is uniform motion a x = 0a x = 0 The y-direction is free fall The y-direction is free fall a y = -ga y = -g The initial velocity can be broken down into its x- and y-components The initial velocity can be broken down into its x- and y-components

Projectile Motion

Projectile Motion at Various Initial Angles Complementary values of the initial angle result in the same range Complementary values of the initial angle result in the same range The heights will be differentThe heights will be different The maximum range occurs at a projection angle of 45 o The maximum range occurs at a projection angle of 45 o

Velocity of the Projectile The velocity of the projectile at any point of its motion is the vector sum of its x and y components at that point The velocity of the projectile at any point of its motion is the vector sum of its x and y components at that point Remember to be careful about the angle’s quadrantRemember to be careful about the angle’s quadrant

Example A cannon fires a cannonball with an initial velocity of 200 m/s at an angle of 40° above ground. How far does the c.b. travel horizontally (range) before it hits the ground?

Example A hose is held 2 m off the ground such that water shoots out horizontally & hits the ground 1.5 m away. What is the initial speed?

Example What is the range of a projectile fired across level ground with a velocity of 100 an angle of 30° above horizontal?