Unit 1
Mechanics

Kinematics - The description of Motion

Topics Covered

• Time, Frequency & Period
• Motion, Distance & Displacement
• Speed & Velocity
• Acceleration

1. Time can be defined as a measure of elapsed regular intervals based on natural recurring events such as the changing of the seasons, sunrise-sunsets, full moon-new moon, high tide and low tide.  The symbol for Time in the metric system is t, and the standard unit of this fundamental scientific quantity is the second (s - for short).  One complete occurrence of a time interval is known as a cycle.  How often cycles are repeated is the frequency of the occurrence of regular events.

2. Frequency, mathematically speaking is the number of repeated actions that occur in a given time. The symbol for Frequency is f and the mathematical formula for frequency is:

The actual unit of frequency is     ("# of cycles" is a counted quantity and not a measured quantity therefore it does not have a unit).  The assigned unit of Frequency in the metric system (SI Units) is the Hertz [Hz - for short].

Example:  What is the frequency of a yo-yo which is being constantly rolled up and down a total of 300 times in 3 minutes?

Solution:

Given: # of cycles = 300; time = 3 min = 3 min  x 60 s/min = 180 s

Find: f          f = cycles/time  = 300 cycles / 180 s = 1.7 H

Period is the reciprocal of frequency.  The period of repeated motion (harmonic) is the time it takes for one complete cycle.  The symbol for period is T (capital letter T) and the unit for period is the second (s).  In equation form we can express period as:

Example:  What is the period of an ocean wave if on average 20 waves are seen striking the shore in 7 seconds.

 Given: # of waves = 20; time = 7 s

Find: T        T= time/cycles = 7 s / 20 waves  = 0.35 s 

 Note that "waves" is a counted quantity therefore it is a  quantity without units.

3. Motion

There are two types of motion:  

• Constant (Uniform) Motion (no acceleration - constant speed) 
• Accelerated Motion (motion at changing speed)

Constant Motion obeys two conditions.  To determine if an object is in constant motion, both these conditions must be met:

1. Motion must be in a straight line
2. Motion must occur at constant speed

Examples:

• The seconds' hand on a clock is moving at constant speed but because it is revolving in a curved path, it cannot be considered constant motion.
• An apple falling from a tree falls in a straight line but because its speed is increasing due to gravitational  acceleration, it is not undergoing constant motion.

Accelerated Motion

1. Motion does not have to be in a straight line
2. The speed of the object changes over time intervals

4. Distance, Displacement, Speed, Velocity

Distance is the change in position from an original point.  The symbol for distance is "d".
Distance is a scalar quantity because it does not take into account direction.

Example:

 If you step out your door and walk on the side walk for 10 m and you turn around and go back home along the same path, you will have traveled a total distance of 20 meters.  Your total displacement will be zero meters.

Displacement is defined as the distance traveled in one direction.  The symbol for displacement is
Displacement is  a vector quantity.  Click on the link Vectors - review to learn more about vectors.
The units of displacement (and of distance) is the meter (m). 

Speed is the rate of change of distance with respect to time. The symbol for speed is   V and the metric base units for speed is the [m/s]

Velocity is the vector equivalent of speed; i.e. the speed an object travels in one particular direction.  The symbol for speed v and the units for speed are " meters per second" or (m/s). 

 The symbol for velocity is . 

To calculate the average speed (or average velocity) of an object we use the general equation  

Example:

If you are walking with your dog in the park at 30 cm/s.  How far will you have traveled in 20 minutes?

Given:  v_av = 30 cm/s = 0.30 m/s
            t _tot =  20 min. = 20 min X 60 s/min = 1200 s

Find: d_tot  -- use the above equation but rearrange it for d_tot

Therefore    d_tot = 0.30 m/s  X 1200 s  = 360 m

5. Acceleration

Acceleration is the rate at which speed increases or decreases.  Acceleration involves a changes in speed or a change in velocity during a time interval. Acceleration can be both a scalar and a vector quantity.  It is a scalar quantity when we consider only its magnitude.  If we also indicate the direction in which an object is accelerating, then we are defining acceleration as a vector quantity.

When an object slows down we say it is decelerating. Deceleration is negative acceleration.  The symbol for acceleration is  a.  The units for acceleration are [m/s² ] .

 Acceleration over a definite  time interval is known as average acceleration.  Acceleration calculated at each instant (very small time segment) along the motion path of an object is known as instantaneous acceleration.

The equation for average acceleration is : 

Note that Dv is the change in speed or velocity during Dt.  This means that if at time t₁ the speed of an object was v₁, at a later time t₂, its corresponding speed is v₂. Also the symbol D indicates the operation "Final - Initial".  

In this case,  Dt = ( t₂ - t₁ )   and   Dv = ( v₂ -v₁)

Example:

A police car is cruising at 50 km/h along a straight road.  A motorist zooms past the cruiser. The police officer  steps on the gas and catches up to the speeder 30 s later.  At that time the police officer notices that her car was going at 80 km/h.  What was the average acceleration during the 30 s?

Given:  v₁ = 50 km/h   v₂ = 80 km/h
            let the initial time t ₁ = 0  then, t ₂ = 30s

Find:  a_av  

The standard unit for acceleration is the m/s² .  Therefore we should convert our answer () to m/s² .

1 km = 1000 m   and 1 h = 3600 s   Therefore:

\  The police cruiser accelerated at 0.28 m/s²

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