Motion in a Straight Line

Motion in a Straight Line

• Physics deals with motion, which is the change in position over time
• Mechanics is the branch of physics about the motion of objects and forces causing it
• Includes kinematics (describing motion)
• Includes dynamics (dealing with the causes of motion/force)

Position

• An object's position is its distance from a reference point
• One direction from the reference point is positive (+), the opposite is negative (-)
• The reference point acts as the origin in a coordinate system

Position vs. Time Graph

• These graphs show an object's position over a period of time
• This allows you to visualize the motion taking place

Distance and Displacement

• Displacement (∆x) is the change in position: ∆x = xf - x0
• The SI unit for displacement is the meter (m)
• Displacement can be positive, negative, or zero
• Distance (d) is the total length of the path traveled by a particle
• Distance is always a positive value, or zero

Question Example: Displacement and Distance

• A man walks from x0 = 2.0m to x1 = 8.0m, then to x2 = 4.0m, then to x3 = 9.0m, and finally to x4 = 5.0m
• To find his displacement (∆x), subtract the initial position from the final position: ∆x = xf - x0 = 5.0m - 2.0m = 3.0m
• To find the distance (d) traveled, add up the distances of each leg of the journey: d = 6.0m + 4.0m + 5.0m + 4.0m = 19.0m

Dimensional Analysis

• Specific symbols are used to denote Dimensions
• Length, [L]
• Mass, [M]
• Time, [T]
• The word "dimension" in physics specifies what a quantity measures
• Length [L] is the dimension for distance
• [L]/[T] is the dimension for speed
• Dimensional analysis can check the correctness of a formula
• Ensures left and right sides of the equation have the same dimensions

Dimensional Analysis Equations

• Speed is [V]=[L]/[T]=[L][T]^-1
• Force is [F]=[M][L]/[T]^2=[M][L][T]^-2
• In these equations, [M], [L], and [T] represent Mass, Length, and Time

Dimensional Analysis Example 1

• To check the equation ∆x = (1/2)at², where ∆x is displacement, a is acceleration and t is time
• [L] = [L] Equation is correct

Dimensional Analysis Example 2

• Check vf = vi + at
• [L]/[T] = [L]/[T] Equation is correct

Vectors

• Magnitude denotes the amount of something and only needs one number, and must be positive, negative, or zero
• Vectors describe the direction a quantity acts in, such as up/down, left/right, etc

Scalars vs. Vectors

• A Scalar quantity that is totally specified by its magnitude, a single number and a unit
• A Vector is a quantity that is specified by both magnitude and direction, needs two or more numbers and a unit
• Scalars include Distance, Speed, Time, Mass and Energy
• Vectors include Displacement, Velocity, Acceleration, Momentum and Force

Components of Vectors

• Vectors are defined by direction (θ) and magnitude |V|
• Magnitude of a vector is its length and is a positive number
• Direction is the angle between the vector and the positive axis
• Range is between 0° and 360° (or 0 and 2π radians)
• Vectors can be:
• Bolded (printed notation)
• Annotated with an arrow (handwritten notation)

Vector Addition

• Two vectors are equal if they have the same magnitude and direction

Multiplication of a Vector by a Scalar

• A unit vector has a magnitude of 1
• The negative of a vector gives zero when added to the original vector
• Multiplying a vector by a positive scalark results in a new vector with the same direction, its magnitude is multiplied by k

Speed and Velocity

• Speed is the rate of change of position
• Speed is a scalar quantity with S.I. unit of m/s
• Average speed is total distance/total time interval defined as Vave=d/∆t=d/(tf-ti)
• Speed has no direction and is always expressed as a positive number or zero (0)

Equation for Average Speed

• To ensure average speed is not negative, Vave = d/|∆t| = d/|tf - ti|

Velocity Defined

• Velocity is speed with direction, can change with changes to speed or direction
• Velocity is a vector quantity and is measured at m/s
• Average velocity is displacement/total time interval
• vx ave = ∆x/∆t = (xf - xi)/(tf - ti)

Velocity Implications

• Velocity has direction and is a positive number (+), negative number (-), or 0 zero (0)

Constant Velocity

• These graphs represent the motion of a particle under constant velocity
• The constant velocity has the same value as the slope of the graph
• The vertical-intercept is labeled as x0
• Blue lines represent faster velocities, Green lines slower velocities
• Black lines represent zero velocity (object is not moving)
• Orange lines represent negative velocity, where the object moves in the −x direction

Average Speed Question

• Find the average speed of a swimmer going across a pool
• Given values: Distance across the pool = 10.0 m in 20.0 s; Swim back 5.00 m in 12.0 s
• To solve: vave = d/|∆t|
• vave = (10.0 m + 5.00 m) / (20.0 s + 12.0 s) = (15.0 m) / (32.0 s)
• Average speed is 0.469 m/s

Average Velocity Question

• Find the average velocity of a swimmer going across a pool
• Given values: Distance across the pool = 10.0 m in 20.0 s; Swim back 5.00 m in 12.0 s
• To solve: vx ave = ∆x/∆t -> [10.0m + (-5.00m)]/(20.0s + 12.0s)
• vx ave ≈ 0.156 m/s

Velocity Example

• Find the velocity of an object traveling 12 m in 4 s
• Vx ave = ∆x/∆t
• Calculated to be 3.00 m/s
• Find how far a train will travel with a velocity of 80 km/hr for 2.5 hr
• ∆x = (Vx ave)∆t
• Calculated as travelling 200km

Motion Measurement

• Calculated by Slope
• Can be used to find (Instantaneous) Velocity

Velocity Over Area

• With the addition of Velocity Over Area, graphs with 3 different velocities can be created
• Instantaneous Velocity

Speed in a graph

• Speed will always measure as a positive, regardless of the direction of motion
• Instantaneous speed

Example question for Velocity and Speed

• Find the Instantaneous Velocity at t = 7.00s
• Find the Instantaneous Speed at t = 7.00s

Average Velocity

• Graph representation
• If tf = 10.0s & ti = 0.00s then Vx ave = (xf-xi)/∆t or (5.00m-0.00m)/(10.0s-0.00s)
• Vx ave = 0.500 m/s

Average Speed

• Average Speed is the Total Distance Traveled divided by the Total Time Interval: Vave = d/∆t
• And if tf = 10.0s & to = 0.00s then Vave = (da+db+dc)/tf-to

Acceleration

• Acceleration is the rate of change of velocity
• Denoted as ax= ∆vx/∆t= (vxf - vxi/tf-ti)
• Velocity has slope, therefore acceleration can be called the (Instantaneous) Acceleration. Is this ax?
• Therefore ax = v2/r?

Acceleration Example

• A car changes speed at a constant rate, from 8.00 m/s /to 20.0 m/s in 4.00s: Solve the two step equation using two different equations? To get to the answer of 3.00 m/s2? ax is Rightward, as the object in question is going from -x to x

Velocity and Acceleration

• Graph for a, b & c
• Velocity, vx(t), versus time, t
• Calculate the three possible values that are achievable

Average Acceleration

• The Acceleration may not remain Constant
• As it relates to this lesson, There are 3 different accelerations
• Δvx=vxf at t= 10.0s, Vx0 at t= 0.00s

Average Acceleration Example

• What is the Average Acceleration, from to = 0.00s until tf = 10.0s?
• Average Acceleration ax ave= -0.600 m/s²

Constant Acceleration

• Includes formulaic equations used for determining constant acceleration

Constant Acceleration Example

• Q: What is the acceleration, if the velocity increases by 3.00 m/s over a time interval of 12.0s, assuming that the acceleration is constant?

Constant Acceleration Formula

• V = v1 - Vi Delta t = 12.0 seconds

Q: A race car accelerates uniformly from 18.5 m/s to 46.1 m/s in 2.7 s.

Determine the acceleration of the car and the car’s displacement during the acceleration.

• Vxf Vxi = axot

Final examples

• Several questions on determining rate, velocity, bike acceleration and travel speeds