**(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

Asked by Abhisek | 1 year ago | 186

**(a)** Let θ be the angle at which the projectile is fired w.r.t the x-axis

θ depends on t

Therefore, tan θ(t) = \( \dfrac{v_x}{v_y}\)

= \(
\dfrac{ (v_{oy} – gt)}{v_{0x}}\)_{ }(since v_{y} = v_{0y} -gt and v_{x} =v_{ox})

θ(t) = tan ^{-1} \(
\dfrac{ (v_{oy} – gt)}{v_{0x}}\)

**(b)** Since, h_{max} =\(
\dfrac{ u^2 sin^2θ}{2g}\) —–(1)

R = \( \dfrac{ u^2 sin^2θ}{g}\)——–(2)

Dividing (1) by (2)

\( \dfrac{h_{max}}{R}\) = \( [\dfrac{ u^2 sin^2θ}{2g}].[\dfrac{ u^2 sin^2θ}{2g}]\)

= tan \( \dfrac{ θ}{4}\)

θ = tan^{-1} (\(
\dfrac{4h_{max}}{R}\))

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.

As a vector is having both direction and magnitude, then is it necessary that if anything is having direction and magnitude it is termed as a vector? The rotation of an object is defined by the angle of rotation about the axis and the direction of rotation of the axis. Will it be a rotation of a vector?