A gun of mass 1000 kg fires a projectile of mass 1 kg with a horizontal velocity of $100~m/s$ The velocity of recoil of the gun in the horizontal direction is:

Question

Accepted Answer

The correct option is (1):

This problem can be solved using the principle of conservation of linear momentum. According to this principle, in an isolated system where no external forces act, the total momentum of the system remains constant. In this scenario, the gun and the projectile form an isolated system. Before firing, both the gun and the projectile are at rest, so the total initial momentum of the system is zero. After firing, the projectile moves forward, and the gun recoils backward to conserve the total momentum.

Let $M$ be the mass of the gun ($1000\ 	ext{kg}$), $m$ be the mass of the projectile ($1\ 	ext{kg}$), $V$ be the recoil velocity of the gun, and $v$ be the velocity of the projectile ($100\ 	ext{m/s}$).

According to the conservation of momentum:
Initial momentum = Final momentum
$0 = MV + mv$

We need to find $V$, the recoil velocity of the gun. Rearranging the equation:
$MV = -mv$
$V = -\frac{mv}{M}$

Now, substitute the given values:
$V = -\frac{(1\ 	ext{kg})(100\ 	ext{m/s})}{1000\ 	ext{kg}}$
$V = -\frac{100\ 	ext{kg}\cdot	ext{m/s}}{1000\ 	ext{kg}}$
$V = -0.1\ 	ext{m/s}$

The negative sign indicates that the recoil velocity of the gun is in the opposite direction to the velocity of the projectile. Therefore, the magnitude of the recoil velocity of the gun is $0.1\ 	ext{m/s}$.

The incorrect option is (2): $5\ 	ext{m/s}$. This value would result if there was a calculation error or a misunderstanding of the momentum conservation principle, leading to an incorrect magnitude.

The incorrect option is (3): $15\ 	ext{m/s}$. This value is significantly higher than the actual recoil velocity and indicates a substantial error in applying the conservation of momentum or a miscalculation.

The incorrect option is (4): $0.5\ 	ext{m/s}$. This option also represents an incorrect calculation. It might arise from a simple arithmetic error in the division, such as dividing by a smaller number or an incorrect multiplication factor.