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KINEMATICS
This is the study of
motion without considering how it is produced. You are going to study motion in a straight with special
attention to SPEED, VELOCITY, and ACCELERATION.
OBJECTIVES. By the end of this topic,
you are expected to be able do the following:
Ø Define distance,
displacement and state the difference between them.
Ø Define the speed of an
object, state the formula for calculating speed and also apply it in answering questions.
Ø Define the velocity of an
object, state the formula for calculating velocity, and apply this formula to
answer questions.
Ø Define the acceleration of
a body, state the formula for calculating it as well as apply it to solve
exercises.
Ø Derive equations of
motions with constant acceleration.
Ø Convert speed in kmh-1
to ms-1 and vice versa using 1kmh-1= 5/18 ms-1
and 1ms-1=18/5kh-1.
Ø Use the correct equation
of motion to solve a given question.
Ø Draw the velocity- time
graph for any given motion.
Ø Use a velocity-time graph
to calculate distance covered over a specific time frame.
Ø Analyse or breakdown, the
shape of a velocity-time graph into simple shapes and find the distance covered
or displacement in a given time interval.
Ø Calculate the acceleration
and deceleration of a particle in a given time after finding the gradient of
the line segment representing the motion.
Ø Find the greatest speed
reached during the motion of a particle.
Ø Convert and use all
equations of motion with constant acceleration to equations of motion under
gravitational force.
Ø Solve problems for
vertical motion under gravity where a=-g for upward movements and a=g for
downward movement
Ø Solve at least 95% of the
problems in Kinematics.
MOTION IN A STRAIGHT LINE.
DISTANCE. This is the space covered by a moving object
from a fixed point. It is denoted by D. When the direction of movement is not
given the space covered remains distance. It is measured centimeters, meter, or
kilometer. Distance only describes magnitude without direction so it is a
scalar quantity.
DISPLACEMENT. This is the space
covered by a moving object from a fixed point in a specific direction. It is
denoted by S. It is measured in centimeters, meters, or kilometers. When distance
and direction are given then displacement is described. Displacement has both
magnitude (distance) and direction, it is a vector quantity.
SPEED. It is defined as the rate at which distance
changes with time. It is denoted by
S and the SI units for speed is cms-1
, ms-1 , or kmh-1. Speed is a scalar quantity.
Speed= Distance (D) ;
Distance=Speed x
Time (T)
Example1. Calculate; a) the
distance in measured in meters travelled in 3 minutes by a body moving with a
constant speed of 15kmh-1, b) the time in seconds, taken by this
body to travel 200m at the same speed.
Solution
a) V=15kmh-1,
t=3m= 3 h
60
Using D=v
D=15km/h x 3 h
60
D= 0.75 km or 750m. The distance travelled is 750m.
b) Using D=v
D=200m, v=15km/h= 15
60
200m= 25 m/s
6
t=48s
The time taken to
travel 200m at 15kmh-1 is 48 s.
CONSTANT SPEED. The statement that the
speed of the car is 20 kilometers per hour (20kmh-1 or 20km/h) means
that if the speed remains unchanged, the car will travel 20km in one hour. The
speed of the car is said to be constant or uniform. It follows that this car
will travel 40km in two hours, 60km in three hours and so on.
VELOCITY. It is defined as the rate
at which displacement(s) changes with time (t).It is denoted by V and the SI
units for measuring it is cms-1, ms-1 or kmh-1.
Velocity is a vector quantity.
Velocity= Displacement(s) ;
Velocity= Displacement x
time
Example 1. At time t=0,a body passes
through a point A and is moving at a constant velocity of 4ms-1.
a) Find how far the body is
from A when t=3s
b) What is the value of t when
the body is 22m from A.
Solution
a) Using displacement=velocity x
t=3s, S=?, V=4ms-1
S=4ms-1
S=12m; displacement= 12metres
b) Velocity is constant
means v=4ms-1
S=22m
t=?
22m=4m/s x t
t=5.5s; time=5.5seconds
The daily usage of speed and velocity are often confused by
many as being same. No! Velocity is speed in a given direction. When direction
is added to speed, we get velocity. As stated above, speed is a scalar quantity
while velocity is a vector quantity.
CONSTANT VELOCITY. Like in speed above the statement that
velocity 20kmh-1 of a car is constant or uniform means that the
velocity remains unchanged.
In practice, the speed and
velocity of a body are seldom constant or uniform. When a car travels 20 in one hour, it
is unlikely that the speed is constant. It is probable that for part of the
time, the car is travelling at more than 20 km in one hour and at some time,
the car’s speed is less than 20km in one hour since the nature of the road and
state of the car are not always perfect. Thus we often refer to the average
speed, or average velocity of a car or body.
Average Speed= Total
distance travelled
Total time taken,
Example.1 A, B, and C are three points in that order, on a straight road
with AB=40km, and BC=90km. A woman travels from A to B at 10kmh-1
and then from B to C at 15kmh-1.Calculate:
a) The time taken to travel
from A to B
b) The time taken to travel from
B to C
c) The average speed of the
woman from A to C
Solution
40km 90km
a) Let D1=AB, D2=BC.
D1=40km, D2= 90km
V1= 10kmh-1
, V2=15kmh-1
Using d=vt from A to B
40km= 10kmh-1
t=4h; time to travel from A to B
b) Using d=vt from B to C
90km= 15km/h x
t=6h; time taken to travel from B to C
c) Average speed= total distance travelled from A to
C
total taken
to move from A to C
Average Speed = 40km +90km
4h + 6h
Average Speed=130km
10h
Average Speed = 13kmh-1 for the whole journey from A to C.
Average Velocity= Total displacement
Total time taken
Example 1.A man walks 400m
due east in a time of 190s, and then 100m due west in a time of 50s.Calculate:
a) His average speed for the
whole journey
b) His average velocity for
the whole journey.
Solution
a) Using average speed= total
distance
Total
time
Average Speed= 400m
+100m
190s + 50s
Average Speed= 500m
240s
Average speed= 2.08 ms-1
b) Average velocity = total
displacement
Time
Average Velocity= 400m
+ (-100m)
190s
+50s
Average Velocity= -300m
=300m East
240s 240 s
Average velocity= 1.25 ms-1
Exercise
1) A, B, and C are three
points lying on a straight road with AB=5km, and BC= 4km. A man runs from A to
B at 20km/h and then walks from B to C
at 8km/h. Find:
a) The total time taken to
travel from A to C (Answer; t=3/4hr or 45mins)
b) The average speed of the
man for the journey from A to C (answer;
12kmh-1)
2) A man walks 150m due north
,in a time of 70s, then 50m due south, in a time of 30s.Find:
a) His average speed (answer;
2ms-1)
b) His average velocity(answer;1ms-1)
3) A car is driven from town A
to town B, 40km away, at an average speed of 60km/h. The car is at B for 10
minutes and then returns back to A.
a) Find the average speed for
the journey from B to A if the average speed for the complete journey is
60km/h. (80km/h)
b) What is the average
velocity of the car for the complete journey? (answer; 0)
4) A ,B, and C are three points lying in that order on a straight
line with AB=60m, and AC=80m.A body moves from A to B at an average speed of
10m/s, then from B to C in a time of 4s, and then returns to B. The average
speed for the whole journey is 5ms-1. Find:
a) The average speed of the
body in the second stage of the motion ,that is from B to C.(5m/s)
b) The average speed of the
body in moving from A to C. (8m/s)
c) The time taken for the
stage of the motion that is from C to B. (10s)
d) The average velocity for
the complete motion, that is from A to C. (3m/s in direction AB)
5) The speed of light is 3x
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