Equations Introduced and Used in this Topic: (All equations can be written and solved as both scalar and vector)

• [latex]a = \dfrac{\Delta v}{t}[/latex]
• [latex]\Delta v = v_f - v_i[/latex]
• [latex]v_f = v_i + at[/latex]
• [latex]d = v_it + \dfrac{1}{2} at^2[/latex]
• [latex]\vec{a} = \dfrac{\vec{\Delta v}}{t}[/latex]
• [latex]v = \dfrac{(v_i + v_f)}{2}[/latex]
• [latex]2ad = v_f^2 - v_i^2[/latex]
• [latex]d = \dfrac{(v_i + v_f) t}{2}[/latex]

Where…

• [latex]v[/latex] is average speed, commonly measured in metres/second (m/s) or kilometres/hour (km/h)
• [latex]\vec{v}[/latex] is average velocity, commonly measured in metres/second (m/s) or kilometres/hour (km/h) and includes a direction
• [latex]v_i[/latex] is initial speed, commonly measured in metres/second (m/s) or kilometres/hour (km/h)
• [latex]\vec{v_i}[/latex] is initial velocity, commonly measured in metres/second (m/s) or kilometres/hour (km/h) and includes a direction
• [latex]v_f[/latex] is final velocity, commonly measured in metres/second (m/s) or kilometres/hour (km/h) and includes a direction
• [latex]\Delta d[/latex] is distance travelled, commonly measured in metres (m), kilometres (km)
• [latex]\vec{\Delta d}[/latex] is change in displacement, commonly measured in metres (m), kilometres (km) and includes a direction
• [latex]a[/latex] is acceleration (deceleration is negative), measured in metres per second squared (m/s²)
• [latex]\vec{a}[/latex] is vector acceleration, measured in metres per second squared (m/s²) and includes a direction
• [latex]t[/latex] is time, commonly measured in seconds (s) or hours (h)

3.1 Average Acceleration

Acceleration (and deceleration) are two commonly experienced phenomenon by anyone who has rode in a vehicle, an elevator or more noticeably so on some ride at an amusement park.  You notice acceleration as a change in either speed or velocity in either speeding up, slowing down or in changing the direction you are traveling. All of these changes to either your speed or velocity means that you are experiencing some sort of acceleration^[1].

Quantifying acceleration and deceleration specifically refers to the rate^[2] of the increase or the decrease in some objects speed (or velocity). This means that:

[latex]a = \dfrac{\Delta v}{t}[/latex] and since [latex]\Delta v = v_f - v_i[/latex], then [latex]a = \dfrac{v_f - v_i}{t}[/latex]

The acceleration equation is written the same for average vector acceleration except it has vector notion written over top of the vector variables.

There are a few distinctions in working with scalar and vector acceleration:

For Scalar Acceleration:

• Acceleration refers to the increase in some objects speed
• Deceleration refers to the decrease in an objects speed (Use a negative for this measure)

For Vector Acceleration:

• Acceleration refers to the change in velocity and can be a positive or a negative value

The units for acceleration are generally given in base SI metric form of m/s²  or as ms^-2.

One can run into the form of units used in the UK that is given as m/s.

One of the rationalizations for using the UK units is due to the mental gymnastics of trying to understand time squared  (s²).

Historically, the concept of acceleration is one that was influenced by observations of the experiments used by various natural philosophers. The oldest surviving records of the study of motion is that of Aristotle (384-322 BCE) who studied the motion of objects falling through water, his medium of choice that he needed to slow down the motion so it could be more easily studied. Water is around 784 times the density of air  which significantly influenced all motions of any falling object.  Any object falling through water reaches a maximum (terminal velocity) in a short order of time which means that it will then fall at a constant velocity downwards. Also, more massive objects will take a longer time to reach terminal velocity when falling downwards.

Aristotle interpreted these observations as follows:

• Heavy objects fall faster than lighter objects.

• The rate at which object fall depended on the density of the medium they were falling through.
• Objects move according to their ratios of their composition of Earth, Water, Air and Fire. All Natural motion therefore occurred to move all elements to their natural place in the universe.

While aware that objects could increase speed, Aristotelian scholars thought that speed was related to the distance separating these four elements and that as the distance decreased the speed increased. To Aristotelians time had no importance in these changes of speed.  Aristotle did distinguish between motion that he considered natural and of motion caused by humans or other non-natural sources that he called violent (unnatural). This interpretation of agents acting on the motion of a body created the foundation of a concept called impetus^[3], where the motion of a body that was thrown would continue in that motion until the impetus initially imparted to it wore off or was exhausted. This concept was refined nearly a millennium later by John Philoponus (490-570) in his modification of the concept of violent motion.

The Middle Ages witnessed the first of the many challenges that grew to eventually overthrow Aristotelian theory. The Persian polymath Avinenna (930-1037) is credited in moving away from pure Aristotelian concepts by claiming that the Philoponean theory of violent motion was motion that was dissipated by external agents such as friction from moving through its medium as the reason for the changes to violent motion of a body.  In this, Avinenna predated Newton in arriving at his first law of motion. Persian science continued to develop concepts related to motion such as Ibn al-Haytham (965-1039) who linked the amount of acceleration related to gravitational acceleration. Abū Rayḥān Muḥammad ibn Aḥmad Al-Bīrūnī (973-1048) and Abu’l-Barakāt al-Baghdādī  (1080–1165) both acted to construct the foundations of what later became know as Newton’s second law of motion that related the external force acting on an object resulted in the object accelerating. History abounds of the great advances coming from Islamic science which not for the total destruction of Baghdad and the House of Wisdom in 1258 during the Mongol invasion by Hulagu Khan could have predated the explosion of European Science by hundreds of years. Instead, from this loss, scientific advancement in Persia slowly diminished in its output and followed what had occurred previously in Egypt and Greece.

European science began to develop in the late Middle Ages from the rediscovery of the works of Aristotle and was well on its path to taking up the reigns left by Persian scientists but was stalled by the Black Death plague in 1348 that killed over one third of the population, reaching higher numbers in urban centres where the heart of science was emerging. It was later in the destruction of the through Byzantine empire by the Ottoman Turks, the infusion of Byzantine and Islamic scientists fleeing to safe refuge, mainly in Italy that the Renaissance started.

Galileo Galilei, perhaps one of the most famous of the Renaissance scientists is credited with accomplishments that he sometimes did not do. For instance, he was claimed to have climbed the Leaning Tower of Pisa and dropped two metal balls both light and heavy and to have proved that both fell at the same rate^[4] and not differently as claimed by Aristotle. The actual experiment is known to have occurred in 1586 by Simon Stevin who dropped two lead spheres, (one that was ten times the weight of the other) from a church tower, 10 metres above ground.

Galileo did however, used an inclined ramp to slow down the acceleration of a falling object and was able to demonstrate that the distance fallen by an object without any friction influencing the fall was proportional to the square of the time the object fell. Specifically:

[latex]d = t^2[/latex]

The following examples all relate to the rate of change in an objects speed or velocity and the acceleration acting on the object. We will be using the following equations for these examples:

• [latex]a = \dfrac{\Delta v}{t}[/latex]
• [latex]\Delta v = v_f - v_i[/latex]
• Which can be expanded to the form:
  [latex]a = \dfrac{v_f - v_i}{t}[/latex]
• [latex]\vec{a} = \dfrac{\vec{\Delta v}}{t}[/latex]
• [latex]\vec{\Delta v} = \vec{v_f} - \vec{v_i}[/latex]
  Which can be expanded to the form:
• [latex]\vec{a} = \dfrac{\vec{v_f} - \vec{v_i}}{t}[/latex]

3.2 Constant Accelerated Motion

Equations Introduced or Used for this Section:

• [latex]a = \dfrac{\Delta v}{t}[/latex]
• [latex]\Delta v = v_f - v_i[/latex]
• [latex]v_f = v_i + at[/latex]
• [latex]d = v_it + \dfrac{1}{2} at^2[/latex]
• [latex]\vec{a} = \dfrac{\vec{\Delta v}}{t}[/latex]
• [latex]v = \dfrac{(v_i + v_f)}{2}[/latex]
• [latex]2ad = v_f^2 - v_i^2[/latex]
• [latex]d = \dfrac{(v_i + v_f) t}{2}[/latex]

The equations you choose to solve an accelerated problem can be found by using the chart shown below.  How we choose which equation to use depends for the most part on which variable is not given or asked for … We can call these variables “not mentioned”.

The chart below indicates which of the four constant acceleration equations you can use for the variable that is not mentioned.

For example, you are given the following data in a problem:

• [latex]a[/latex] = a (constant) value
• [latex]v_i[/latex] = a value
• [latex]v_f[/latex] = a value
• [latex]d[/latex] = asked to find
• [latex]t[/latex] = not mentioned

The equation to use is:

[latex]2ad = v_f^2 - v_i^2[/latex]

In a setting where there is a constant average acceleration, it is possible to combine several equations together to create four accelerated motion equations. Specifically:

• [latex]v_f=v_i+at[/latex]
• [latex]2ad = v_f^2 - v_i^2[/latex]
• [latex]d = v_it + \dfrac{1}{2} at^2[/latex]
• [latex]d = \dfrac{(v_i + v_f) t}{2}[/latex]

It is quite easy to show how two of these four equations are derived …

[latex]d = \dfrac{(v_i + v_f) t}{2}[/latex] is simply the expansion of [latex]d=vt[/latex] (originally given as [latex]v=\dfrac{d}{t}[/latex]) where the speed of the object has been replaced by taking the average of the initial and final speed (or velocity [latex]= \dfrac{(v_i + v_f) }{2}[/latex]

In this case you can see that:

For the equation [latex]d=vt[/latex] when you replace [latex]= \dfrac{(v_i + v_f) }{2}[/latex], you are left with [latex]d = \dfrac{(v_i + v_f) t}{2}[/latex]

[latex]v_f=v_i+at[/latex] is simply the rearrangement of the basic acceleration equation [latex]a=\Delta \dfrac{v}{t}[/latex] where [latex]\Delta v[/latex] is replaced with [latex]v_f-v_i[/latex] and [latex]a=\Delta \dfrac{v}{t}[/latex] is rewritten as [latex]at=\Delta v[/latex]

When combined you are left with [latex]at=v_f-v_i[/latex] more commonly written as [latex]v_f=v_i+at[/latex]

The other two equations [latex]d = v_it + \dfrac{1}{2} at^2[/latex] and [latex]2ad = v_f^2 - v_i^2[/latex] require a little more work to derive.

NOTE: For these four equations to work correctly, the acceleration must be uniform or constant.

The above examples all relate to the constant acceleration of a body in a straight line.  In all cases the speed of the object will be changing. It is also possible to have an object experience constant acceleration but not have any change in speed. This occurs for an object traveling in a circular path, such as a planet orbiting the Sun or a satellite orbiting a planet.

All motion depends on the direction of the acceleration that is acting on the body. If the acceleration is in the same direction (or opposite to the direction) that the body moves then the speed will be changing. If this acceleration is acting at 90° to the direction of motion, then the object will move in a circular path, remaining at the exact same speed but constantly changing its velocity.  Motion in a circular path will be studied much later in this text.

Exercise Answers

3.1 Average Acceleration

1. 5.0 m/s²
2. 2.9 m/s²
3. 29 m/s
4. 1.7 s
5. 11 s
6. 5.1 s
7. 2 g’s
8. 2.21 × 10¹⁶ m/s²
9. 1.9 s

3.2 Accelerated Motion

1. 300 m/s²
   1. 1.25 m (≈ 1.2 m)
   2. 5.0 m/s
   1. 4.5 s
   2. 47 m 4
   1. 37.5 m (≈ 37 m)
   2. 113 m  (≈ 110 m)
2. 90 m
3. 29 m/s
4. 5.0 m/s², deceleration = 8.0 s to stop
5. 3.0 s
6. 46 m/s
7. 5.0 m/s²
   1. 2.2 m/s²
   2. 83 m
8. 73 m
9. 153 m or 150 m
10. 20.8 m or 21 m

3.3 Accelerated Motion

1. 94 m
2. 1.3 m
3. Granny makes it!!! (by 0.35 s)
4. 32 m/s

3.4.1 ICBC Road Sense Campaign

1. Cars deceleration = 1.1 g’s

3.4.2 Star Trek Discovery Episode “The Vulcan Hello”

1. 62 m/s² or 6.3 g’s

Media Attributions

• “Focus Accident” by Johnathan Nightingale is licensed under CC BY-SA 2.0 licence.