Motion Notes

Motion

Distance

Total length of a path, regardless of direction.
If the path is linear, displacement and distance are the same.
Measurements:
- meters (m)
- centimeters (cm)
- kilometers (km)
- millimeters (mm)

Displacement

Difference between the initial and current position of an object.
A vector quantity, possessing both magnitude and direction.
If the path involves changes in direction, displacement will be less than the distance.
Measurements:
- meters (m)
- centimeters (cm)
- kilometers (km)
- millimeters (mm)

Distance & Displacement Examples

Walking from home to school and back (250 meters apart):
- Distance = $250m + 250m = 500m$
- Displacement = $0m$ (since the final position is the same as the initial position)
Driving 3.0 km north and then 3.0 km west:
- This example would require the Pythagorean theorem to solve for displacement.

Speed

Magnitude (size) of the rate of change of position of an object. It is directionless.
Speed is equal to the ratio of change in distance in time.
Units:
- meters per second (m/s)
- kilometers per hour (km/h)
Formula:
- $V = \frac{d}{t}$
- Where:
  - D = distance/displacement (meters, m)
  - T = time (seconds, s)
  - V = speed/velocity (m/s)

Velocity

Rate of change of the position of an object in the direction of its motion.
Magnitude and direction define velocity.
Velocity is equal to the ratio of change in displacement to change in time.
Units: meter per second (m/s)
Formula:
- $V = \frac{d}{t}$
Example: A car travels 61 km/hr for 200 seconds. Determine the distance traveled in km.
*Another Example: A person walks 100 meters south and then 100 meters east in one minute. Determine the velocity (in m/s).

Average Speed & Velocity

A constant velocity is when the magnitude of the speed and the direction of motion are unchanged over a period of time.
The average speed is the total distance travelled by an object divided by the time in motion.
Example: Determine the average speed of an object that travels 200 meters in 10 seconds and then 100 meters in the subsequent 20 seconds.

Acceleration

Acceleration is a vector defined as the rate of change of velocity of an object over time.
The units are meters per second squared ( $m/s^2$ ).
Objects can accelerate in different ways:
- Increase in speed (linear acceleration)
- Change in direction (radial acceleration)
- Decrease in speed (deceleration)
Formula:
- $a = \frac{v - u}{t}$
- or
- $a = \frac{V<em>f - V</em>i}{t}$
- Where:
  - a = acceleration
  - v = final velocity
  - u = initial velocity
  - t = time (seconds)
Examples of acceleration scenarios:
- Going onto the highway (positive acceleration, positive velocity)
- Reversing into a parking spot (negative acceleration, negative velocity)
- Coming to a stop sign (negative acceleration, positive velocity)
- Reversing out of the driveway (positive acceleration, negative velocity)

Understanding Speed & Acceleration

	Positive Acceleration	Negative Acceleration
Positive Velocity	Speeding up / going forwards	Slowing down / going forwards
Negative Velocity	Slowing down / going backwards	Speeding up / going backwards

If \Delta v = v - u > 0, then the change in velocity, and thus acceleration, is positive.
If \Delta v = v - u < 0, then the change in velocity, and thus acceleration, is negative.
If |v| > |u|, then the magnitude of the final speed is greater than the magnitude of the initial speed, and so the object has sped up.
If |v| < |u|, then the magnitude of the final speed is less than the magnitude of the initial speed, and so the object has slowed down.
Examples:
- A vehicle changes velocity from 25 m/s to 9 m/s.
- A vehicle changes velocity from -3 m/s to -12 m/s.
- A vehicle changes velocity from -16 m/s to -2 m/s.

Gravity

Gravity is the force by which a planet or body draws objects towards its center.
Anything with mass has gravity.

Acceleration due to Earth’s Gravitational Field

Earth’s mass causes a gravitational force on objects, the force is strongest at Earth’s surface.
Acceleration of approximately $9.81 m/s^2$ .
All objects in free fall will hit the ground at the same time regardless of mass.
Examples:
- What is the velocity of a quarter dropped from a tower after 10 seconds? Answer: -98.1 m/s
- How long will it take an object that falls from rest to attain a velocity of 147 m/s? Answer: 15 s
- If a ball that is freely falling has attained a velocity of -19.6 m/s after two seconds, what is its velocity five seconds later? Answer: -68.6 m/s

Graphing Motion at Constant Speed

Position-Time Graph:
- The slope of the graph represents the change of displacement over time.
- The direction of the slope indicates the direction of travel.
- A flat graph indicates the object is not moving.
- A linear graph indicates the object is moving at a constant speed.

Understanding Position-Time Graphs

Slope of position-time graph = velocity over that interval of time
The steeper the slope, the higher the speed (value of velocity)
Slope is zero --> velocity is zero (object at rest)
Slope is positive --> velocity is constant, positive
Slope is negative --> velocity is constant, negative
Slope is a curve --> velocity is not constant (object accelerating)

Position Graph Meaning

Above Axis: Ahead of the starting point
Below Axis: Behind the starting point
Flat: Not moving
Positive Slope: Going forwards
Negative Slope: Going backwards
Linear: At constant velocity
Example Problems:
- How far did the vehicle travel in 4 seconds?
- What was the average speed of the vehicle over 4 seconds?
- How long was the vehicle stationary?
- What was the total distance of the journey?
Example Problems:
- What is Carol’s displacement after 6 seconds?
- What is Carol’s velocity?
- If Carol continues to jog at the same velocity how far will she have travelled after 22 seconds?
What is the velocity between each of these points? O - A, A - B, B - C

Understanding Velocity-Time Graphs

Velocity vs. Time Graph
Velocity-Time graphs show the change of velocity over an elapsed time.
Time is always the independent variable.
Velocity is always the dependent variable.

Velocity-Time Graph (Slope)

The slope of a Velocity-Time graph is equal to acceleration.
Slope = change in velocity / time

Speed/Time Graph

These objects are moving with a positive velocity.
These objects are moving with a negative velocity.
Acceleration is constant, positive; speeding up
Acceleration is constant, negative; slowing down

Graph & Meaning

Graph	Meaning
Flat	Constant speed
Above Zero	Going forwards
Below Zero	Going backwards

Displacement = Area
Direction = Whether it's above or below the time axis

How to find Displacement?

Distance is area under velocity vs. time graph.

Changing Motion

A velocity graph could be used to represent changing motion.
To determine the average speed and velocity one must determine the total distance and displacement of the travelling object using the area of the graph.

Graphing Accelerated Motion

Position Graphs
- The slope of the tangent line to the graph represents the instantaneous velocity.
- The direction of the graph indicates the direction of motion (forwards or backwards).
- The shape of the graph represents the acceleration.
  - A linear graph means a = 0 (constant velocity).
  - A curvy parabolic graph indicates acceleration.

Accelerated Motion

Positive Tangent Slope: Going forwards
Negative Tangent Slope: Going backwards
Linear: At constant velocity
Curving: Accelerating (or decelerating)

Acceleration Graphs

Acceleration versus Time
Shows how velocity changes as time passes, either positively or negatively.
The area between the graph and time axis represents the change in velocity as time passes. *The graph does not indicate whether it is accelerating / decelerating -- that depends on whether the velocity is positive or negative.

Acceleration Graph Meaning

Flat: No motion OR constant speed
Above Zero: Positive velocity
Below Zero: Negative velocity
Ex:
Change in Velocity = Area
Area = 2 x 15 = +30 m/s
Could be speeding up or slowing down