State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful :

(a) Addition of any two scalars

(b) Adding a scalar to a vector which has the same dimensions

(c) Multiplying a vector by any scalar

(d) Multiplying any two scalars

(e) Adding any two vectors

(f) Addition of a vector component to the same vector.

Asked by Pragya Singh | 1 year ago | 68

**(a) Meaningful:**

The addition of two scalar quantities is meaningful only if they both represent the same physical quantity.

**(b) Not Meaningful:**

The addition of a vector quantity with a scalar quantity is not meaningful

**(c) Meaningful:**

A scalar can be multiplied with a vector. For example, force is multiplied with time to give impulse.

**(d) Meaningful:**

A scalar, irrespective of the physical quantity it represents, can be multiplied by another scalar having the same or different dimensions.

**(e) Meaningful:**

The addition of two vector quantities is meaningful only if they both represent the same physical quantity.

**(f) Meaningful:**

A component of a vector can be added to the same vector as they both have the same dimensions.

Answered by Abhisek | 1 year ago**(a) **Show that for a projectile the angle between the velocity and the x-axis as a function of time is given by

\(\theta (t)=tan^{-1}(\dfrac{v_{0y-gt}}{v_{ox}})\)

**(b)** Shows that the projection angle θ_{0} for a projectile launched from the origin is given by

\(\theta_{0}=tan^{-1}(\dfrac{4h_{m}}{R})\)

where the symbols have their usual meaning

A cyclist is riding with a speed of 27 km/h. As he approaches a circular turn on the road of a radius of 80 m, he applies brakes and reduces his speed at the constant rate of 0.50 m/s every second. What is the magnitude and direction of the net acceleration of the cyclist on the circular turn?

A fighter plane flying horizontally at an altitude of 1.5 km with a speed of 720 km/h passes directly overhead an anti-aircraft gun. At what angle from the vertical should the gun be fired for the shell with muzzle speed 600 m s^{-1} to hit the plane? At what minimum altitude should the pilot fly the plane to avoid being hit? (Take g = 10 m s^{-2} ).

A bullet fired at an angle of 30° with the horizontal hits the ground 3.0 km away. By adjusting its angle of projection, can one hope to hit a target 5.0 km away? Assume the muzzle speed to be fixed, and neglect air resistance.

Can we associate a vector with

**(i)** a sphere

**(ii)** the length of a wire bent into a loop

**(iii)** a plane area

Clarify for the same.