Class 12 Mathematics Dynamics Solutions has been updated according to the latest syllabus of 2080. It means the notes of dynamics chapter provided in this article contains all the new exercise that has recently been updated. Now you don’t need to go anywhere searching for the notes of this chapter because we are here to serve you.

## Chapter – 19

__Dynamics__

In Class 12 mathematics, the chapter on dynamics is focused on understanding the motion of objects and the forces that influence this motion. It involves using calculus, particularly differentiation and integration, to analyze how particles move under various forces.

This chapter also delves into relative motion, examining how the motion of objects is perceived in different frames of reference, particularly in terms of velocity and acceleration.

## Class 12 Mathematics Dynamics Solutions PDF

This PDF will provide the solutions of every question from the 4th exercise of class 12 dynamics chapter. If you want the notes of other exercises then you can choose from the button given above.

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### Introduction: Projectiles

An object, thrown vertically upwards with a given velocity, begins to fall vertically downwards after reaching the highest point (i.e. after the velocity has reduced to zero). If the object is supposed to be near the surface of the earth and the resistance due to air is neglected the acceleration due to gravity is then almost constant. It is directed vertically downwards and its magnitude is g = 9.8ms² (or 10ms² approximately).

If the object is thrown obliquely (i.e. at an angle α to the horizon) with a certain velocity near the surface of the earth, it moves along a curved path. At any point on the path, the only effective acceleration is the acceleration due to gravity. As said above, it is almost constant and directed downwards in this case also.

### Motion of Projectiles

A projectile is any object having a given initial velocity and moving along a path determined by the force of gravity and frictional resistance of the air. In an idealized situation, the object is assumed so small that it may be regarded as a particle. The may be considered to be moving in vacuum. Lastly the particle is supposed to be close to the surface of the earth so that the acceleration due to gravity remains almost constant. In our further discussion, unless otherwise stated, we consider only such an idealized model of physical situation.

A particle thrown obliquely near the surface of the earth is called a projectile, the path along which the projectile moves, is called its trajectory. The angle a which the direction of projection makes in the plane of projection with the horizontal plane through the point of projection is called the angle of projection; the time taken by the particle to come back to thehorizontal plane again is called the time of flight; and the distance between the point of projection and the point where the particle strikes the horizontal plane again is called the range or horizontal range.

Let a particle be projected from a point O with an initial velocity u and making an angle a with the horizontal plane through the point of projection O. Further, suppose v is the velocity of the projectile at some point P on the trajectory OAB and inclined to the horizontal OB at an angle 0.

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