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In the previous chapter, we have analyzed objects with constant velocity. When the velocity of an object changes, the object is said to be accelerated. In this chapter, we will further analyze the motion and think about how fast an object moves, how far it moves and for how long.
 
1. What is acceleration?
2. Average Acceleration and Instantaneous Acceleration
3. More Velocity-time Graph
4. Displacement and Constant Acceleration
5. Acceleration Due to Gravity
6. Chapter 3 Quiz
  Section Section 1. What is acceleration?
 
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When an object's velocity changes, it accelerates. Acceleration shows the change in velocity in a unit time. Velocity is measured in meters per second, m/s, so acceleration is measured in (m/s)/s, or m/s2, which can be both positive and negative.

 
 
  Section Section 2. Average Acceleration and Instantaneous Acceleration

The average acceleration is the ratio between the change in velocity and the time interval.
 
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Formula for Acceleration

For example, if a car moves from the rest to 5 m/s in 5 seconds, its average acceleration is
5 m/s divided by 5 s

An instantaneous acceleration is the change in velocity at one moment. We will study instantaneous acceleration more in depth later in the chapter.
     
   
QUESTION: If a car accelerates from 5 m/s to 15 m/s in 2 seconds, what is the car's average acceleration? 
m/s/s 

QUESTION: How long does it take to accelerate an object from rest to 10 m/s if the acceleration was 2 m/s2? 
s 

QUESTION: Carl started to run at 10 km/h when he left his house. He arrived at school 30 minutes later. How fast was he running when he arrived there? Assume that his average acceleration was 30 km/h2. 
km/h 

 
 
  Section Section 3. More Velocity-time Graph

Let's examine more about a velocity-time graph. We can say that the tangent of a velocity-time graph represents instantaneous acceleration since
 
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Formula for getting acceleration from a velocity-time graph

For example, the instantaneous acceleration at t = 3 s in the graph below is 3 m/s2, since the graph has a slope of 3 at t = 3 s.
A velocity-time graph

     
   
QUESTION: What is the instantaneous acceleration of the above object at t = 0? 
m/s 

QUESTION: What is the average acceleration of the whole trip? (When t = 7, velocity = 26 m/s) 
m/s/s 

QUESTION: Does above object has a constant acceleration? 
(Yes/No) 

 
 
  Section Section 4. Displacement and Constant Acceleration

In this section, we will assume that acceleration is always constant.

We know that the area under the line of a velocity-time graph represents the displacement. Therefore, the equation
 
Check!
Formula for calculating displacement from initial and final velocity

is true, where Vi is the initial velocity and Vf is the final velocity, since the area of a triangle is 1/2 * width * height.

A velocity-time graph

     
   
QUESTION: If a car accelerated from 5 m/s to 25 m/s in 10 seconds, how far will it travel?
m 
     
   

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The final velocity of a uniformly accelerated object is

Formula for calculating the final velocity from initial velocity, acceleration, and time, 

where:

  • Vf is the final velocity in m/s, 
  • Vi is the initial velocity in m/s, 
  • a is acceleration in m/s2, and 
  • t is time in seconds.
Therefore, by substituting it into the previous equation,
 
Check!
Formula for calculating displacent from initial velocity, acceleration, and time is true.
     
   
QUESTION: What is the displacement of a car whose initial velocity is 5 m/s and then accelerated 2 m/s2 for 10 seconds? 
m 
     
    From equations Formula for calculating displacement from initial and final velocity and Formula for calculating the final velocity from initial velocity, time, and acceleration, we can also say that
Derivation
 
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Therefore, Formula illustrating the relationship between initial and final velocity, acceleration, and displacement is true. These red equations are very important and you should be very familiar with them. (It doesn't mean that you should memorize these formulae. Learn by using them.)
     
   
QUESTION: What is the final velocity of a car that accelerated 10 m/s2 from rest and traveled 180m?
m/s 

 
 
Section

Section 5. Acceleration Due to Gravity

Galileo was the first to find out that all objects falling to Earth have a constant acceleration of 9.80 m/s2 regardless of their mass. Acceleration due to gravity is given a symbol g, which equals to 9.80 m/s2.

Therefore, if you drop a pen, it should behave like this...
 
Time (s)  Velocity (m/s)  Displacement (m)
0 0 0
1 9.8 4.9
2 19.6 19.6
3 29.4 44.1
4 39.2 78.4

For all previous equations, we can substitute g for a:
 
Check!
Formula for calculating the final velocity from initial velocity, gravity, and time

Formula that illustrate the relationship between initial and final velocity, gravity, and displacement

Formula for calculating displacement from initial velocity, time, and gravity

     
   
QUESTION: How long will it take for an apple falling from a 29.4m-tall tree to hit the ground? 
s 

 
 
  Section

Section 6. Chapter 3 Quiz 

Take the Chapter 3 Quiz and see how well you can do!

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