Physical quantities are broadly classified into vectors and scalars. Physical quantities which have a sense of direction are known as vectors, these are categorised as polar and axial. Vectors cannot be added by simple laws of algebra but by adopting vector algebra. Resultant of two or more vectors could be zero depending upon their magnitude and direction. When resultant of three vectors are zero or null then Lami's theorem is applicable. Vector substraction is similar to addition where vector addition of one vector and negative vector of other is done. In our daily life vector addition and vector substraction is very important as far as application of mechanical force is concerned.
(i) Define vector physical quantity. Give example.
(ii) Distinguish between Polar and Axial vectors. Give example
(iii) What are unit vector and null vector?
(iv) State Lami's theorem of equilibrium of three vectors.
(v) State triangle law of vector addition.

An aeroplane is flying with velocity \(\vec{v}_{p}\) (= 100 ms^-1) towards east, with respect to ground through motionless air and \(\vec{v}_{w}\) is the wind velocity with respect to ground. The total velocity of aeroplane is \(\vec{v}=\overrightarrow{v_{p}}+\overrightarrow{v_{w}}\). The magnitude of the velocity is often called speed. The heading of the plane is the direction in which the nose of the plane points. Infact, it is the direction in which the engine propels the plane. Resultant velocity and direction is the vector sum of different components of velocity, while relative velocity is the velocity of one with respect to another obtained by vector substraction.
(i) What is resultant velocity of plane if the wind velocity is towards south with respect to ground at 25 ms^-1?
(ii) Define relative velocity? How it is determined?
(iii) If the wind blows with velocity 25 ms^-1 northwards, then by which angle the plane velocity is deflected from east?
(iv) Vectors cannot be added algebraically. Why?
(v) Mention the condition when the magnitude of vectors \(\overline{\mathbf{A}}-\overline{\mathbf{B}}\)and \(\overline{\mathbf{A}}+\overline{\mathbf{B}}\) is same.

Projectile is the name given to a body thrown with some initial velocity with the horizontal direction and then allowed to move in two dimensions under the action of gravity alone, without being propelled by any engine or fuel. A projectile moves under the combined effect of two velocities: one a uniform velocity in the horizontal direction and other a uniformly changing velocity. It is observed that path of projectile which is known as trajectory is parabolic in shape.
(i) Does the time of flight, horizontal range and maximum height depend on the mass of the projectile?
(ii) State the relation between the maximum height attained by the projectile and the maximum range.
(iii) What is the angle between velocity and acceleration at the highest point of projectile path?
(iv) A body is projected with velocity u at angle e with the horizontal, what would be the angle and speed of projectile when it strike at the ground at same horizontal plane from which it is projected?
(v) What are the angles of projection for same initial velocity of projection at which horizontal range of the projectile is same?
(vi) Write the equation of the path of projectile, projected at angle e with initial velocity u, from a horizontal plane.

If a body moves in circle with a constant speed, then the magnitude of the velocity is constant but the direction of the velocity vector is changing all the time. Thusvelocity is changing with time. Therefore the motion of the body is accelerated. The acceleration is directed towards the centre of the circle and is called centripetal acceleration a_c. If m is the mass of the body then the centripetal force is F_c = ma_c·
(i) What is effect on the linear momentum, angular momentum and kinetic energy of body, moving uniformly in circular path?
(ii) What is the magnitude and direction of a_c of body moving in circular path of radius r with speed v?
(iii) A stone of mass m is tied to a string is whirled in circular path of radius r in horizontal plane at a constant speed v. What is the angle between velocity and acceleration? Also determine the tension in the string.
(iv) State the relation between linear velocity (v) and angular velocity (ω) of body moving in circular path of radius r,
(v) What would be the direction of motion of stone if at certain time its string get broken?
(vi) In a non-uniform circular motion with tangential acceleration a_t and radial acceleration a_r. What would be the resultant acceleration?