Chapter "1" Lesson "2.b"

2.1 Motion study "b"

 Acceleration 

In tracing the motion of Mr Samer's car, we notice that the speed of car does not remain uniform throughout the journey . We say that the Car is either accelerating (i.e. getting faster) or decelerating (i.e. getting slower). 

Acceleration is defined as the rate of change of velocity.

The SI unit of acceleration is meter per square second (m/s2).

Mathematically, it is represented by the formula:

Acceleration = change in velocity (m/s)

                               Time taken (s)

                     = Final velocity - Initial velocity

                                    Time Taken

Uniform Acceleration

The acceleration of an object is uniform when the rate of change of its velocity remains Constant. For example, a uniform acceleration of 2 m/s2 means that the velocity of the object changes y 2 m/s every one Second later.

Pause & think

  An Object moves in a circular path, but its speed is uniform. Does it have an acceleration?

Let's consider the motion of a car accelerating at 2 m/s2. If the Car starts from rest, its velocity = 0 m/s at time t=0 s. One Second later, its velocity increase to 2 m/s During the next one second, its velocity will have increased another 2 m/s to become 4 m/s. At the end of the third second, its velocity is 6 m/s. At each one second time interval, its velocity increases at the Same amount of 2 m/s every one second. We say that the car accelerates uniformly. It has uniform acceleration.

Worked Example 1.2

A car accelerates uniformly from rest to a velocity of 15 m/s in 3 s.

(a) Calculate the acceleration of the car

(b) Predict the value of the velocity of the car at the end of 5 s if its acceleration remains constant. Explain your prediction. 

Solution

(a) As the car starts from rest, its initial velocity = 0 m/s.

acceleration = Final velocity - initial velocity

                                time taken

                   = 15 - 0  = 5 m/s2

                           3

(b) The acceleration of the car is 5 m/s2, it means that every 1 s later its velocity increases by 5 m/s. Hence, the velocity of the car at the end of 5 s is ( 15 + 5+5) = 25 m/s.

Alternatively, We Know from (a), its initial velocity is 0 m/s and its acceleration is 5 m/s2; at the end of 5 s, its velocity can be calculated using the formula

 acceleration = velocity / time taken

velocity = 5 × 5 = 25 m/s 

Test yourself  "on Facebook"

A skate boarder slides down a slope from rest with a uniform acceleration of 3 m/s2. What is his speed at the end of 4 s?

Speed-Time Graph

On another occasion, Bara observes the speedometer in his father's car at 1-minute intervals and records the speed "v" against time "t" in the table below. This data traces the journey from his home to the traffic junction just before the car hits the highway.

We Can easily  translate the information above into a graph of speed "v" against time "t" to show the motion of the Car.

During the first 2 minutes, i.e. from t = 0 min. to t = 2 min, the car accelerates uniformly. The speed of the car increases at a constant rate of 40 km/h every 1 minute. Then the speed of the car becomes uniform at 80 km/h during the next 3 minutes, from t = 2 min. to t = 5 min. During this period, the speed of the car doesn't change. The acceleration of the car is zero.

At t = 5 min, the car accelerates again for the next 2 minutes. However this time the car accelerates non-uniformly as the increase in speed is not constant. At the end of t =7 min, the Car slows down or decelerates and finally comes to rest after 8 minutes into its journey. 

= Final speed - Initial Speed = Acceleration

            Time interval

Area under Speed- Time Graph

suppose Bara' father passes the traffic junction, and is able to accelerate uniformly to a speed of 36 km/h, or 10 m/s in 4 sm and maintains this uniform speed of 10 m/s for the next 4 s before bringing the car to rest uniformly.

**Note: Unit conversion 36 km/h = 3600 m  = 1 m/s

                                                       3600 s 

We can represent this part of the journey by the speed - time graph below

To obtain the total distance traveled   by the car during this journey, we calculate the area under the speed -time graph;

Total distance traveled = Area "A" + Area "B" + Area "C" = Area of Trapezium

                                   = 0.5 * (10+4) * 10

                                   =  70 m

Acceleration of Speed - Time Motion

To obtain the acceleration "a" of the car during the first four seconds and the last two seconds, we calculate the slopes under the graph.

a (t = 0 to t = 4) = slope of positive slope = 10 ÷ 4 = 2.5 m/s2

a (t = 8 to t = 10) = slope of negative slope = -10 ÷ 2 = -5 m/s2

chapter "1" , Lesson # 2.a

1.2 Motion study

Let's follow Mr Samer's car one morning and trace his journey as he drives his son Bara' to his school.

Bara' checks his digital Watch which registers 06:10 a.m. when Mr Samer starts his car engine. Let's Call this time t = 0 s,  Bara' watches the speedometer carfully at each minute interval.

Distance-time graph 

we can translate the information gathered by Bara' into a table of distance travelled (d), by the car against time (t), and Plot distance-time graph, as shown below

The distance-time graph of the journey 

As observed from the distance-time graph, Mr Samer does not drive at a constant or uniform speed. In the first minute, from t= 0 min to t=1 min, the distance covered by the car is 0.20 km, which gives the average speed of the car as 0.020 km/ min. The distance covered during the second minute is 0.30 km, giving the average speed during the second minute as 0.30 km/ min.

  

Average speed( t=0 min to t =1min = 0) = 0.20 /1 = 0.20 km/h

Average speed( t=1 min to t =1min = 2) = 0.30 /1 = 0.30 km/h

It Can be observed from the distance-time graph that the slope of the curve between t = 0 min to t = 2 min increases and it represent the " instantaneous speed" during this period. Hence the car's speed is increasing. We say that the car is accelerating. 

However, during the next four minutes, from t = 2 min to t= 6 min, the distance covered by the car is uniform or constant at 0.50 km for every one minute. Hence the car travels at a uniform speed of 0.50 km/min during this period;

Slope(t =2-3min) = (1-0.5)/(3-2) = 0.5 /1 = 0.5 km/min (average speed)

Slope(t =5-6min) = (2.5-2)/(6-5) = 0.5 /1 = 0.5 km/min (average speed)

The uniform speed from t = 2 min to t=6 min is demonstrated by a straight line with uniform slope "gradient".

Uniform speed (t=2 min to t= 6 min) = slope "gradient" (t=2 min to t= 6 min)
                                                        = 2.5 - 0.5  =  0.2    = 0.5 km /min

                                                                6-2           4

During the next 3 minutes from t = 6 min.to t = 9 min. the Car slows down as the distance covered by the car each minute later decreases is from 0.3 km to 0.05 km, the speed of the car decreases from 0.3 km /min to 0.05 km/ min, The slope "gradient" of the distance - time graph shows corresponding decrease.

During the last one minute of its journey the car is at rest as the distance traveled is zero. the slope "gradient" of distance - time graph is zero.

Graphical summary of the motion:

*** Test yourself "on Facebook"

Sketch the distance- time graph of a body that is travelling 
(a) at constant speed,
(b) with increasing speed; and
(c) with decreasing speed

chapter"1" Kinematics, Lesson #1

what is Kinematics? 

Kinematics is the branch of physics which involves description of motion without examining the forces which produce it. It is a study of how changes in motion are linked to time. 

 1.1 Speed and velocity

Speed is a scalar quantity. It tells us how fast or how slow a body is moving, It describes how the distance of the object changes with time.

A jumbo jet can fly at a speed of 1000 km/h, i.e. it can travel a distance of 1000 km in 1 hour!

However, a common snail crawls at an average speed of 0.002 km/h. That's right! it covers a mere distance of 2 meters in 1 hour!

Velocity is a vector quantity. It also tells us how fast or how slow a body is moving. But as a vector quantity, velocity is defined by a magnitude as well as a direction.So velocity is speed in a specific direction.

    

To investigate the difference between speed and velocity, watch this funny video:

http://www.youtube.com/watch?v=DRb5PSxJerM

Average Speed

Due to traffic Conditions, it is unlikely that a moving body will be able to maintain a constant or uniform speed throughout its journey. Hence it is more appropriate to use the term average speed.

The SI unit of average speed is meter per second (m/s).

A more common and convenient unit of speed for our vehicle is  kilometer per hour (km/h)

Worked Example 1.1

Laith begins cycling from home and reaches his school 12 km away in 30 minutes. Calculate his average cycling speed in (a) km/h;   (b) m/s

Solution
(a) average speed = Total distance / total time = 12 km/0.5 h = 24 km/h
(b) 24 km /h = 24 000 m/3600 s = 7 m/s

What will you be able to do at the end of this chapter "Kinematics"?

1. state what is meant by speed and velocity

2. calculate average speed using distance traveled / time taken;

3. state what is meant by uniform acceleration and calculate the value of an acceleration;

4. interpret given examples of non-uniform acceleration;

5. Plot and interpret a distance-time and speed time graph;

6. deduce from the shape of distance-time graph when a body is (i) at rest    (ii) moving with uniform speed  (iii) moving with non-uniform speed;

7. deduce from the speed - time graph when a body is

 (i) at rest,

(ii) moving with uniform speed,

(iii) moving with uniform acceleration,

(iv) moving with non-uniform acceleration;

8. Calculate the area under a speed-time graph to determine the distance travelled for motion with uniform speed or uniform acceleration; and

    9. state the acceleration of free fall for a body near to the earth is constant and approximately 10 m/s2.