08 Equations of Motion (Updated).pdf presentation

Equation of motion

1. GEG 124: ENGINEERING
APPLIED MATHEMATICS
II
DERIVATIONS OF EQUATIONS OF MOTION
BY ANALYTICAL METHOD
MODULE 4

2. Intended Outcome
At the end of this module, students are expected to:
1. understand the derivation of first equation of motion
2 . understand the derivation of second equation of motion
3. . understand the derivation of third equation of motion
4. solve questions involving (1) to (3).

3. Equations of motion
There are three equations of motion
First equation - v= u+ at
Second equation - S=푢?+
1
2
�?
2
Third equation - �
2
=푢
2
+2�?

4. Relation among velocity, distance, time and acceleration
is called equations of motion.

First Equation of Motion:
The final velocity (v) of a moving object with uniform
acceleration (a) after time, t.
Let
The initial velocity = u.
Final velocity = v
Time = t
Acceleration = a

5. First Equation of Motion

6. Second Equation of Motion

8. Third Equation of Motion

10. Expression for the distance travelled by body in
nth seconds of its motion

12. Motion under gravity

14. Sign Convention (Cartesian)

15. Example 1

16. Solution

17. Example 2

18. Solution

19. Example 3

21. Example 4
A particle starts from rest and moves with a uniform acceleration 1.2 �/?
2
. What is its
velocity after 20 seconds and how far has it moved in this time?
Solution:
�=1.2 �/?
2
, ?=20?푒??, 푢=0, �= ?
Using Equation (1)� = 푢 + �?
� = 0 + 1.2(20)=24�
And �=푢? + 
1
2
�?
2
=0 + 
1
2
(1.2)(20)
2
=240 �

22. Example 5
The speed of a particle increases uniformly from 12�/? to 16�/?. The total
distance covered during the process is 96�. Find:
a.The acceleration of the particle.
b.How long this takes the particle.
Solution:
�=12�/?,  �=16�/?,  ?=96�
Using �
2 =푢
2
+2�?
16
2
=12
2
+2�(96)
�=0.58
�
?
2
And � = 푢 + �?
16 = 12 + 0.58(?) ?= 6.9 ?.

23. Example 6
A train travelling at 27푘�/ℎ is accelerated at the rate of 0.5�/?. Find the distance
covered by the car in 20 ?푒?�???.
Solution:
�=27푘�/ℎ =7.5�/?
Using � = 푢? + 
1
2
�?
2
 =7.5(20)+ 
1
2
(0.5)(20)
2
=250 �

24. Example 7
A burglar’s car had a start with an acceleration of 2�/?
2
. A police vigilant Patrol came after 5
seconds and continued to chase the bungler’s car with a uniform velocity of 20�/?. Find the
time taken, in which the police van will overtake the burglar’s car.
Solution:
Burglar’s car
푢=0,  � = 2�/s
2
�
1 = distance travelled at ?+5
�
1=푢?+
1
2
�?
2
�
1=0+
1
2
(2)(?+5)
2
=(?+5)
2
Therefore at meeting point (?+5)
2 = 20?
  ?
2 + 10? + 25 − 20? = 0
?
2
−10? + 25= 0 ⇒   ? = 5?

25. Practice Questions
1. The speed limit of a particular section of freeway is 25 m/s. The right travel lane is
connected to an exit ramp with a short auxiliary lane. Cars would have a comfortable
deceleration of −2.0 m/s
2 for 3.0 s in the auxiliary lane if they were driving at the speed limit.
a. What speed will cars have when they are done decelerating in this way? (This is
also the speed limit of the exit ramp.)
b. What minimum length should the auxiliary lane be to allow for this deceleration?
Drivers don't always drive at the speed limit, and highway engineers take this into
consideration.
c. Assume a car could decelerate at four times the "comfortable" rate without losing
control. At what maximum speed could a car enter an auxiliary lane with the length
you calculated in part b. and still exit at the intended speed?
d. Assume a driver was traveling on the freeway at the speed you calculated in part c.
What distance is needed to decelerate this car to the speed limit of the exit ramp at
the "comfortable" rate ?

27. 2. A car with an initial velocity of 60 mph needs 144 feet to come to a complete stop.
Determine the stopping distance of this same car with an initial velocity of…
(a.) 30 mph (b) 20 mph (c) 10 mph
Note: The rate of change of velocity is not affected by inital velocity in this problem.
Fast cars and slow cars slow down at the same rate.
3. A typical commercial jet airliner needs to reach a speed of 180 knots before it can
take off. (A knot is a nautical mile per hour and is nearly equal to half a meter per
second.) If such a plane spends 30 s on the runway estimate…
a) its acceleration.
b) the minimum runway length.

28. 4. A 10 car subway train is sitting in a station. It reaches its cruising speed after
accelerating at 0.75 m/s
2 for distance equivalent to the length of the station (184 m). It
then travels at a constant speed towards the next station 18 blocks away (1425 m).
a) Determine the train's cruising speed.
b) Determine the time it took for the train to accelerate from rest to its
cruising speed.
c) How long does it take the train to travel the 18 blocks to the next station?
The driver stops the train in the second station in half the distance it took to start it at
the first station.
d) What is deceleration of the train in the second station?