a 1.75kg pistol lies at rest on an essentially frictionless bench. it accidentally fires and shoots a 25 gram bullet parallel to the bench. How far has the bullet moved by the time the gun has recoiled 2.60mm?
To solve this problem, we need to apply the principle of conservation of momentum. According to this principle, the total momentum of a system before and after an event remains constant, provided no external forces act on it.
The momentum of an object is defined as the product of its mass and velocity: momentum = mass × velocity.
Let's assume the initial velocity of the pistol and the bullet is zero. After the bullet is fired, it acquires a velocity, v, and the pistol recoils with a velocity, V.
Using the conservation of momentum, we can set up an equation:
(initial momentum) = (final momentum)
Since the pistol is initially at rest, its momentum is zero. The bullet's momentum is given by (mass of the bullet × velocity of the bullet). The pistol's momentum after firing is (mass of the pistol × velocity of the pistol).
So, our equation becomes:
(0) = (mass of the bullet × velocity of the bullet) + (mass of the pistol × velocity of the pistol)
Now, we can substitute the given values into the equation. The mass of the bullet is 25 grams, which is equal to 0.025 kg. The mass of the pistol is 1.75 kg. The velocity of the pistol is zero (initially), and the bullet's velocity is what we are trying to find. The recoil of the pistol is given as 2.60 mm, which we need to convert to meters (2.60 mm = 0.0026 m).
Therefore, our equation becomes:
0 = (0.025 kg × velocity of the bullet) + (1.75 kg × (-0.0026 m/s))
Simplifying further, we get:
0 = 0.025 kg × velocity of the bullet - 1.75 kg × 0.0026 m/s
0.0445 kg·m/s = 0.025 kg × velocity of the bullet
Now, we can solve for the velocity of the bullet:
velocity of the bullet = 0.0445 kg·m/s / 0.025 kg
velocity of the bullet = 1.78 m/s
So, the bullet has a velocity of 1.78 m/s.
Finally, to find how far the bullet has moved, we need to calculate the distance traveled in the time it took to recoil 2.60 mm. Since the bullet's velocity is constant, we can use the formula:
distance = velocity × time
The time can be calculated by dividing the recoil distance by the velocity of the pistol:
time = recoil distance / velocity of the pistol
time = 0.0026 m / 0.0026 m/s
time = 1 second
Therefore, the distance traveled by the bullet is:
distance = velocity of the bullet × time
distance = 1.78 m/s × 1 s
distance = 1.78 meters
Hence, the bullet has moved approximately 1.78 meters by the time the gun has recoiled 2.60 mm.