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Topic 4 Uniform Acceleration

- Chapter 2 Motion in One Dimension

Quantities in Motion

- Any motion involves three concepts
- Displacement
- Velocity
- Acceleration
- These concepts can be used to study objects in

motion

2.1 Displacement

- Defined as the change in position
- f stands for final and i stands for initial
- May be represented as ?y if vertical
- Units are meters (m) in SI, centimeters (cm) in

cgs or feet (ft) in US Customary

Displacements

Speed

- The average speed of an object is defined as the

total distance traveled divided by the total time

elapsed - Speed is a scalar quantity

Speed, cont

- Average speed totally ignores any variations in

the objects actual motion during the trip - The total distance and the total time are all

that is important - SI units are m/s

2.2 Velocity

- It takes time for an object to undergo a

displacement - The average velocity is rate at which the

displacement occurs - generally use a time interval, so let ti 0

Speed vs. Velocity

- Cars on both paths have the same average velocity

since they had the same displacement in the same

time interval - The car on the blue path will have a greater

average speed since the distance it traveled is

larger

Average Velocity, Constant

- The straight line indicates constant velocity
- The slope of the line is the value of the average

velocity

Average Velocity, Non Constant

- The motion is non-constant velocity
- The average velocity is the slope of the blue

line joining two points

Uniform Velocity

- Uniform velocity is constant velocity
- The instantaneous velocities are always the same
- All the instantaneous velocities will also equal

the average velocity

2.3 Acceleration

- Changing velocity (non-uniform) means an

acceleration is present - Acceleration is the rate of change of the

velocity - Units are m/s² (SI), cm/s² (cgs), and ft/s² (US

Cust)

Average Acceleration

- Vector quantity
- When the sign of the velocity and the

acceleration are the same (either positive or

negative), then the speed is increasing - When the sign of the velocity and the

acceleration are in the opposite directions, the

speed is decreasing

Average Acceleration

2.4 Relationship Between Acceleration and Velocity

- Uniform velocity (shown by red arrows maintaining

the same size) - Acceleration equals zero

Relationship Between Velocity and Acceleration

- Velocity and acceleration are in the same

direction - Acceleration is uniform (blue arrows maintain the

same length) - Velocity is increasing (red arrows are getting

longer) - Positive velocity and positive acceleration

Relationship Between Velocity and Acceleration

- Acceleration and velocity are in opposite

directions - Acceleration is uniform (blue arrows maintain the

same length) - Velocity is decreasing (red arrows are getting

shorter) - Velocity is positive and acceleration is negative

2.5 Kinematic Equations

- Used in situations with uniform acceleration

Graphical Interpretation of the Equation

Problem-Solving Hints

- Read the problem
- Draw a diagram
- Choose a coordinate system, label initial and

final points, indicate a positive direction for

velocities and accelerations - Label all quantities, be sure all the units are

consistent - Convert if necessary
- Choose the appropriate kinematic equation

Problem-Solving Hints, cont

- Solve for the unknowns
- You may have to solve two equations for two

unknowns - Check your results
- Estimate and compare
- Check units

2.6 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 objects

original motion - All objects falling near the earths surface fall

with a constant acceleration - The acceleration is called the acceleration due

to gravity, and indicated by g

Acceleration due to Gravity

- Symbolized by g
- g 9.80 m/s²
- When estimating, use g 10 m/s2
- g is always directed downward
- toward the center of the earth
- Ignoring air resistance and assuming g doesnt

vary with altitude over short vertical distances,

free fall is constantly accelerated motion

Free Fall an object dropped

- Initial velocity is zero
- Let up be positive
- Use the kinematic equations
- Generally use y instead of x since vertical
- Acceleration is g -9.80 m/s2

vo 0 a g

Free Fall an object thrown downward

- a g -9.80 m/s2
- Initial velocity ? 0
- With upward being positive, initial velocity will

be negative

Free Fall -- object thrown upward

- Initial velocity is upward, so positive
- The instantaneous velocity at the maximum height

is zero - a g -9.80 m/s2 everywhere in the motion

v 0

Thrown upward, cont.

- The motion may be symmetrical
- Then tup tdown
- Then v -vo
- The motion may not be symmetrical
- Break the motion into various parts
- Generally up and down

Non-symmetrical Free Fall

- Need to divide the motion into segments
- Possibilities include
- Upward and downward portions
- The symmetrical portion back to the release point

and then the non-symmetrical portion

3.3 Motion in Two Dimensions

- Using or signs is not always sufficient to

fully describe motion in more than one dimension - Vectors can be used to more fully describe motion
- Still interested in displacement, velocity, and

acceleration

Displacement

- The position of an object is described by its

position vector, - The displacement of the object is defined as the

change in its position

Velocity

- The average velocity is the ratio of the

displacement to the time interval for the

displacement - The instantaneous velocity is the limit of the

average velocity as ?t approaches zero - The direction of the instantaneous velocity is

along a line that is tangent to the path of the

particle and in the direction of motion

Acceleration

- The average acceleration is defined as the rate

at which the velocity changes - The instantaneous acceleration is the limit of

the average acceleration as ?t approaches zero

Unit Summary (SI)

- Displacement
- m
- Average velocity and instantaneous velocity
- m/s
- Average acceleration and instantaneous

acceleration - m/s2

Ways an Object Might Accelerate

- The magnitude of the velocity (the speed) can

change - The direction of the velocity can change
- Even though the magnitude is constant
- Both the magnitude and the direction can change

3.4 Projectile Motion

- An object may move in both the x and y directions

simultaneously - It moves in two dimensions
- The form of two dimensional motion we will deal

with is called projectile motion

Assumptions of Projectile Motion

- We may ignore air friction
- We may ignore the rotation of the earth
- With these assumptions, an object in projectile

motion will follow a parabolic path

Rules of Projectile Motion

- The x- and y-directions of motion are completely

independent of each other - The x-direction is uniform motion
- ax 0
- The y-direction is free fall
- ay -g
- 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 - The heights will be different
- The maximum range occurs at a projection angle of

45o

Some Details About the Rules

- x-direction
- ax 0
- x vxot
- This is the only operative equation in the

x-direction since there is uniform velocity in

that direction

More Details About the Rules

- y-direction
- free fall problem
- a -g
- take the positive direction as upward
- uniformly accelerated motion, so the motion

equations all hold

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 - Remember to be careful about the angles quadrant

Problem-Solving Strategy

- Select a coordinate system and sketch the path of

the projectile - Include initial and final positions, velocities,

and accelerations - Resolve the initial velocity into x- and

y-components - Treat the horizontal and vertical motions

independently

Problem-Solving Strategy, cont

- Follow the techniques for solving problems with

constant velocity to analyze the horizontal

motion of the projectile - Follow the techniques for solving problems with

constant acceleration to analyze the vertical

motion of the projectile

- An object may be fired horizontally
- The initial velocity is all in the x-direction
- vo vx and vy 0
- All the general rules of projectile motion apply

- Follow the general rules for projectile motion
- Break the y-direction into parts
- up and down
- symmetrical back to initial height and then the

rest of the height