Fundamental Problem 15.12
The cannon and support without a projectile have a mass of 250 kg. If a 20 kg projectile is fired from the cannon with a velocity of 400 m/s , measured relative to the cannon, determine the speed of the projectile as it leaves the barrel of the cannon. Neglect rolling resistance.
To determine the speed of the projectile as it leaves the barrel of the cannon, we need to apply the principle of conservation of momentum. According to this principle, the total momentum before the projectile is fired is equal to the total momentum after the projectile is fired.
Before the projectile is fired, the cannon, support, and projectile are at rest. The total initial momentum is therefore zero.
After the projectile is fired, the cannon and support gain momentum in the opposite direction to maintain momentum conservation. Let's denote the speed of the cannon and support after the projectile is fired as Vc.
The total final momentum can be expressed as the sum of the momentum of the cannon and support (250 kg * Vc) and the momentum of the projectile (20 kg * Vp). Since we know the velocity of the projectile relative to the cannon (400 m/s), we can express the speed of the projectile as the sum of Vc and Vp: Vc + Vp.
Applying the conservation of momentum, we have:
0 = (250 kg * Vc) + (20 kg * Vp)
Solving for Vp, we get:
20 kg * Vp = -250 kg * Vc
Vp = (-250 kg * Vc) / 20 kg
Now, we can substitute the values into the equation. Since the problem statement does not provide the value of Vc, it cannot be determined without additional information.
To determine the speed of the projectile as it leaves the barrel of the cannon, we can use the principles of conservation of momentum.
The initial momentum of the cannon and support without a projectile is zero, as it is at rest. The final momentum of the system will be the momentum of the cannon, support, and the projectile combined.
The momentum of an object can be determined using the equation: momentum = mass * velocity.
Let's denote the speed of the projectile as it leaves the barrel as v. We can set up the conservation of momentum equation:
(initial momentum) = (final momentum)
(0) = (mass of cannon and support) * (final velocity of cannon and support) + (mass of projectile) * (final velocity of projectile)
Since we are neglecting rolling resistance, the final velocity of the cannon and support will be the same as the initial velocity, which is zero.
(0) = (250 kg) * (0) + (20 kg) * (v)
0 = 0 + (20 kg) * (v)
We can now solve for v:
0 = 20 kg * v
0 = v
Therefore, the speed of the projectile as it leaves the barrel of the cannon is 0 m/s.