Which of the given relations are true for any arbitrary motion in space?

**(a)** \(v_{average} = \left (d\frac{1}{2} \right )\left ( v\left ( t_{1} \right ) + v\left ( t_{2} \right ) \right )\)

**(b)** \(v_{average} = \dfrac{\left [ r\left ( t_{2} \right ) – r\left ( t_{1} \right )\right ]}{\left ( t_{2} – t_{1}\right )}\)

**(c)** \(v(t) = v\left ( 0 \right ) + at\)

**(d)** \(r(t) = r(0) + v(0)t + \left (\dfrac{1}{2} \right )at^{2}\)

**(e)** \(a_{average} = \dfrac{\left [ v\left ( t_{2} \right ) – v\left ( t_{1} \right )\right ]}{\left ( t_{2} – t_{1}\right )}\)

Asked by Abhisek | 1 year ago | 65

**Right answer is (a) False**

**Explanation:-**

It is given that the motion of the particle is arbitrary. Therefore, the average velocity of the particle cannot be given by this equation.

**Right answer is (b) True**

**Explanation:-**

The arbitrary motion of the particle can be represented by this equation.

**Right answer is (c) False**

**Explanation:-**

The motion of the particle is arbitrary. The acceleration of the particle may also be non-uniform. Hence, this equation cannot represent the motion of the particle in space.

**Right answer is (d) False**

**Explanation:-**

The motion of the particle is arbitrary; acceleration of the particle may also be non-uniform. Hence, this equation cannot represent the motion of a particle in space.

**Right answer is (e) True**

**Explanation:-**

The arbitrary motion of the particle can be represented by this equation.

Answered by Pragya Singh | 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.