Physics

Q.6. Let $A=\left[-1,1\right]$ , Then, discuss whether the following functions defined on  $A$ are one-one, onto, or bijective:

(i)  $f\left(x\right)=\frac{x}{2}$

(ii)  $g\left(x\right)=\left|x\right|$

(iii)  $h\left(x\right)=x\left|x\right|$

(iv)  $k\left(x\right)=x^2$ .

Q.6. Let $A=\left[-1,1\right]$ , Then, discuss whether the following functions defined on  $A$ are one-one, onto, or bijective:

(i)  $f\left(x\right)=\frac{x}{2}$

(ii)  $g\left(x\right)=\left|x\right|$

(iii)  $h\left(x\right)=x\left|x\right|$

(iv)  $k\left(x\right)=x^2$ .

4.32

(a) Show that for a projectile the angle between the velocity and the x-axis as a function

of time is given by

(b) Shows that the projection angle _0 for a projectile launched from the origin is

given by

where the symbols have their usual meaning.

4.31 A cyclist is riding with a speed of 27 km/h. As he approaches a circular turn on the road of radius 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?

4.30 A fighter plane flying horizontally at an altitude of 1.5 km with speed 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 ).

4.29 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.

4.28 Can you associate vectors with

(a) The length of a wire bent into a loop

(b) A plane area

(c) A sphere?

Explain.