State the principle of conservation of momentum - Leaving Cert Physics - Question a - 2007

A rocket rises due to the conservation of momentum when it expels gas downwards at high speed. By Newton's third law, this action creates an equal and opposite reaction. As the rocket expels gas (which has momentum), the momentum of the rocket increases in the opposite direction (upwards), allowing it to ascend. The effective upward thrust is the result of the downward momentum of the exhaust gases.

To calculate the initial momentum of trolley A, we use the formula for momentum:

$p = mu$

Here, the mass $m$ of trolley A is 12 kg and its initial velocity $u$ is 3.5 m s⁻¹. Therefore, the initial momentum of trolley A is:

$p_A = 12 ext{ kg} imes 3.5 ext{ m s}^{-1} = 42 ext{ kg m s}^{-1}$

After the collision, the two trolleys stick together and move with a common velocity, $v$. To find this velocity, we use the conservation of momentum. The initial momentum of the system (which is only from trolley A, since trolley B is at rest) must equal the final momentum:

$p_{initial} = p_{final}$

Before the collision:

$p_{initial} = p_A = 42 ext{ kg m s}^{-1}$

After the collision, if both trolleys (combined mass $m = 12 ext{ kg} + 12 ext{ kg} = 24 ext{ kg}$) move together with velocity $v$:

$p_{final} = mv = 24 ext{ kg} imes v$

Setting these equal:

$42 = 24v$

Therefore, solving for $v$ gives:

v = rac{42}{24} = 1.75 ext{ m s}^{-1}