The height y and the distance \(x\) along the vertical plane of a projectile on a certain planet (with no surrounding atmosphere) are given by \(𝑦 = ( 8 𝑡 − 5t^ 2 )\) meter and \(x = 6t\) meter, where \(t\) is in seconds. The velocity with which the projectile is projected is:

1. \(8\ \text{m/s}\)

2. \(6\ \text{m/s}\)

3. \(10\ \text{m/s}\)

4. Not obtainable from the data

The range of a particle when launched at an angle of \(15^°\) with the horizontal is \(1.5\ \text{km}\). What is the range of the projectile when launched at an angle of \(45^°\) to the horizontal:

1. \(1.5\ \text{km}\)

2. \(3.0\ \text{km}\)

3. \(6.0\ \text{km}\)

4. \(0.75\ \text{km}\)

A projectile thrown with a speed \(v\) at an angle \(θ\) has a range \(R\) on the surface of the Earth. For same \(v\) and \(θ\), its range on the surface of the Moon will be \(\left(\text{acceleration due to gravity on moon}=\dfrac g6\right)\):

1. \(R/6\)

2. \(6 R\)

3. \(R/36\)

4. \(36 R\)

A ball is projected with kinetic energy \(E\) at an angle of \(45^°\) to the horizontal. At the highest point during its flight, its kinetic energy will be:

1. Zero

2. \(\dfrac E2\)

3. \(\dfrac E{\sqrt2}\)

4. \(E\)

At the top of the trajectory of a projectile, the magnitude of the acceleration is:

1. Maximum

2. Minimum

3. Zero

4. \(g\)

A body is projected at such an angle that the horizontal range is three times the greatest height. The angle of projection is:

1. \(25^\circ 8 '\)

2. \(33^\circ 7 '\)

3. \(42^\circ 8 '\)

4. \(53^\circ 8 '\)

Two bodies are projected with the same velocity. If one is projected at an angle of \(30^°\) and the other at an angle of \(60^°\) to the horizontal, the ratio of the maximum heights reached is:

1. \(3 : 1\)

2. \(1 : 3\)

3. \(1 : 2\)

4.  \( 2 : 1\)

If the range of a gun that fires a shell with muzzle speed v is R, then the angle of elevation of the gun is 

(1) ${\cos }^{-1}\left(\frac{{\displaystyle v^2}}{{\displaystyle Rg}}\right)$

(2) ${\cos }^{-1}\left(\frac{{\displaystyle gR}}{{\displaystyle v^2}}\right)$

(3) $\frac{1}{2}\left(\frac{{\displaystyle v^2}}{{\displaystyle Rg}}\right)$

(4) $\frac{1}{2}{\sin }^{-1}\left(\frac{{\displaystyle gR}}{{\displaystyle v^2}}\right)$

If a body \(A\) of mass \(M\) is thrown with a velocity \(v\) at an angle of \(30^°\) to the horizontal and another body \(B\) of the same mass is thrown with the same speed at an angle of \(60^°\) to the horizontal. The ratio of the horizontal range of \(A\) to \(B\) will be:

1. \(1 : 3\)

2. \(1 : 1\)

3. \(1 : \sqrt3\)

4. \( \sqrt3:1\)

Four bodies \(P\), \(Q\), \(R\) and \(S\) are projected with equal velocities having angles of projection \(15^{\circ},\) \(30^{\circ},\)\(45^{\circ},\) and \(60^{\circ}\) with the horizontal respectively. The body having the shortest range is?