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STUDY MATERIAL FOR BSC MATHEMATICS
DYNAMICS
SEMESTER - VI, ACADEMIC YEAR 2020 – 21
UNIT CONTENT PAGE Nr
I PROJECTILES 02
II COLLISION OF ELASTIC BODIES 17
III SIMPLE HARMONIC MOTION 25
IV MOTION UNDER THE ACTION OF CENTRAL FORCE 33
V DIFFERENTIAL EQUATIONS OF CENTRAL ORBITS 47
Page 1 of 56
STUDY MATERIAL FOR BSC MATHEMATICS
DYNAMICS
SEMESTER - VI, ACADEMIC YEAR 2020 – 21
UNIT - I
PROJECTILES
Definition:
1. The angle of projection is the angle that the direction in which the particle is initially
projected makes with the horizontal plane through the point of projection
2. The velocity of projection is the velocity with which the particle is projected
3. The trajectory is the path which the particle describes
4. The range on a plane through the point of projection and the point where the trajectory
meets that plane
5. The time of flight is the interval of time that elapse from the instant of projection till the
instant when the particle again meets the horizontal plane through the point of
projection.
The path of a projectile:
Let a particle be projected from O with a velocity u at an angle ߙ to the horizon.
Take O as the origin, the horizontal and upward vertical through O as axes of
ݔ ܽ݊݀ ݕ respectively.
The initial velocity u can be split into two components, which are u cosߙ in the
horizontal direction and using in the vertical direction
The horizontal component u cosߙ is constant throughout the motion as there is no
horizontal acceleration.
The vertical component using ߙ is subject to an acceleration g down wards.
Let p (ݔ ,ݕ) be the position of the particle at time t secs after projection .
Then ݔ = horizontal distance described in t seconds
= (u cosߙ).t …………..①
Y= Vertical distance described in t sec
ଵ ଶ
= (u sin ߙ).t - ݃ݐ …………..②
ଶ
మ
௫ ଵ ௫
From ① t= - ݃
௨ ௦ఈ ଶ ௨మ௦మఈ
మ
௫
ݕ ൌ ݔ ݐܽ݊ ߙ െ ………………③
ଶ௨మ௦మఈ
ଶ ଶ ଶ ଶ ଶ
ൌ>2ݑ ܿݏ ߙݕ ൌݔ ݐܽ݊ ߙ.2ݑ ܿݏ ߙെ݃ݔ
మ మ మ
ଶ௨ ௦ ఈ௬ ଶ௨ ି௦ ఈ ௦ ఈ௫ ଶ
= ൌ - ݔ
Page 2 of 56
STUDY MATERIAL FOR BSC MATHEMATICS
DYNAMICS
SEMESTER - VI, ACADEMIC YEAR 2020 – 21
మ మ మ
ଶ ଶ௨ ି௦ ఈ ௦ ఈ ௫ ଶ௨ ௦ ఈ
= > ݔ െ ² = ݕ
ቀ ቁ
మ ర మ మ
௨ ି௦ ఈ ௦ ఈ ௨ ି௦²ఈ ௦² ఈ ିଶ௨ ௦ ఈ
= > ݔ െ ² - = ݕ
ቀ ቁ
²
మ మ
ି௨ ି ௦² ఈ ି௨ ି ௦² ఈ
= ݔ ² ݕ
ቀ ቁ ቀ ቁ
ଶ
Transfer the origin to the point
మ మ
௨ ି௦ ఈ ௦ ఈ ௨ ௦² ఈ
= , ,
ቀ ቁ
ଶ
మ
ଶ ିଶ௨ ି ௦² ఈ
Then ݔ = , y …….④
మ
ଶ௨ ௦² ఈ
④ is the equaƟon of the paral with latusrectam ,
Whose axis is vertical and downwards and where vertex
మ మ
௨ ି௦ ఈ ௦ ఈ ௨ ௦² ఈ
1’s the point = ,
ቀ ቁ
ଶ
మ
ଶ௨ ௦² ఈ
Latus rectum m =
ଶ ଶ
= (ucosߙሻ
ଶ
= x square of the horizontal velocity.
Characteristic of the motion of a projectile:
1. Greatest height attained by a projectile:
Page 3 of 56
STUDY MATERIAL FOR BSC MATHEMATICS
DYNAMICS
SEMESTER - VI, ACADEMIC YEAR 2020 – 21
At A, the highest point the particle will be moving only horizontally ,
having lost all of its vertical verlocity.
Let AB = h = the great at height reached.
ଶ ଶ
Initial upward vertical velocity = u sin ߙ (ݒ ൌݑ 2ܽݏሻ
The acceleration = -g
The final vertical velocity = 0
ଶ
Hence 0 =(u sin ߙሻ +2(-g)h.
ଶ ଶ
= >ݑ ݏ݅ ߙ ൌ2݄݃
మ
௨ ௦² ఈ
h=
ଶ
i.e) the vertex of the parabola is two highest point of the path
2. Time taken to reach the greatest height:
Let T be the time from 0 to A .At this time initial velocity u sin ߙ is
reduced to zero.
௩ୀ௨ା௧
Hence O =u sin ߙ݃
୳ ୱ୧୬ ఈ
= > T =
1. Time of flight:
When the particle arrives at c, the effective vertical distance it has
described is zero.
ଵ ଶ
∴=ut + ܽݐ
ଶ
Page 4 of 56
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