A 2500kg gun fires a 10.0kg projectile with a muzzle speed of 550m/s.
A) determine the initial recoil speed of the gun
B) If the recoil is against a constant resisting force of 4200N, find the time taken to bring the gun to rest.
To answer these questions, we can use the principle of conservation of momentum. According to this principle, the total momentum before the event (in this case, firing the gun) is equal to the total momentum after the event.
Let's start with part A:
A) To find the initial recoil speed of the gun, we need to consider the total momentum before the gun is fired. The total momentum is given by the product of the mass and the velocity. The mass of the gun is 2500 kg, and the muzzle speed of the projectile is 550 m/s. Since the gun and projectile are initially at rest, the initial momentum is zero. Therefore, we have:
Total momentum before the event = Total momentum after the event
0 = (mass of gun) × (recoil velocity of gun) + (mass of projectile) × (final velocity of projectile)
Rearranging the equation, we can solve for the recoil velocity of the gun:
(recoil velocity of gun) = - ((mass of projectile) × (final velocity of projectile)) / (mass of gun)
Plugging in the values:
(recoil velocity of gun) = - ((10.0 kg) × (550 m/s)) / (2500 kg)
Calculating this, we find that the recoil velocity of the gun is approximately -2.2 m/s. Note that the negative sign indicates that the gun moves in the opposite direction of the projectile.
Now let's move on to part B:
B) We are given that the recoil is against a constant resisting force of 4200 N. To find the time taken to bring the gun to rest, we can use Newton's second law, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration:
Force = mass × acceleration
In this case, the force acting on the gun is the resisting force, which is 4200 N. The mass of the gun is still 2500 kg. The acceleration can be calculated using the equation:
Acceleration = (change in velocity) / (time taken)
Since we are looking for the time taken to bring the gun to rest, the final velocity of the gun is 0 m/s. So, the change in velocity is equal to the initial recoil velocity of the gun (-2.2 m/s). Rearranging the equation, we get:
(time taken) = (change in velocity) / (acceleration)
Plugging in the values:
(time taken) = (-2.2 m/s) / (4200 N / 2500 kg)
Calculating this, we find that the time taken to bring the gun to rest is approximately 0.0013 seconds.