A rifle with a weight of 17 N fires a 3.6 g bullet
with a speed of 231 m/s.
The acceleration of gravity is 9.81 m/s
2
.
Find the recoil speed of the rifle.
Answer in units of m/s.
To find the recoil speed of the rifle, we need to apply the principle of conservation of momentum. The momentum before firing is equal to the momentum after firing.
The momentum of an object is given by the formula:
Momentum = mass × velocity
Given that the rifle has a weight of 17 N, we can convert this into mass using Newton's second law:
Weight = mass × acceleration due to gravity
Therefore, mass = Weight / acceleration due to gravity
Mass = 17 N / 9.81 m/s^2
Now we have the mass of the rifle.
Next, we can calculate the momentum before firing by multiplying the mass of the rifle by its initial velocity, which is zero because the rifle is at rest:
Momentum before firing = Mass of the rifle × Initial velocity of the rifle
The momentum after firing is given by the mass of the bullet multiplied by its velocity:
Momentum after firing = Mass of the bullet × Velocity of the bullet
According to the principle of conservation of momentum, the momentum before firing is equal to the momentum after firing:
Momentum before firing = Momentum after firing
So we can set up the equation:
Mass of the rifle × Initial velocity of the rifle = Mass of the bullet × Velocity of the bullet
Now we can solve for the initial velocity of the rifle:
Initial velocity of the rifle = (Mass of the bullet × Velocity of the bullet) / Mass of the rifle
Substituting the given values:
Initial velocity of the rifle = (3.6 g × 231 m/s) / (17 N / 9.81 m/s^2)
Remembering to convert grams to kilograms (1 g = 1 × 10^-3 kg), we get:
Initial velocity of the rifle = (3.6 × 10^-3 kg × 231 m/s) / (17 N / 9.81 m/s^2)
Calculating this, we find the initial velocity of the rifle.
Since the rifle is initially at rest, the recoil speed of the rifle is equal in magnitude but opposite in direction to the initial velocity:
Recoil speed of the rifle = -Initial velocity of the rifle
Therefore, the recoil speed of the rifle can be found by taking the negative value of the initial velocity of the rifle.
Performing the necessary calculations, we get the recoil speed of the rifle in units of m/s.
To find the recoil speed of the rifle, we can use the concept of conservation of momentum.
The momentum of an object is given by the equation:
momentum = mass * velocity
In this case, the momentum before the bullet is fired is equal to the momentum after the bullet is fired. The initial momentum is zero since the rifle is initially at rest. The final momentum is given by the product of the mass of the bullet and its velocity.
Let's calculate the initial and final momentum:
Initial momentum (before firing the bullet) = 0
Final momentum (after firing the bullet) = mass of bullet * velocity of bullet
Given:
mass of the bullet = 3.6 g = 0.0036 kg
velocity of the bullet = 231 m/s
Final momentum = 0.0036 kg * 231 m/s
Now, since momentum is conserved, the change in momentum of the rifle is equal in magnitude but opposite in direction to the change in momentum of the bullet. Therefore:
Change in momentum of rifle = - Final momentum
The change in momentum of the rifle can be calculated using the equation:
Change in momentum = mass of the rifle * change in velocity
Here, the mass of the rifle is not given, but we can use the weight of the rifle to find the mass using the equation:
Weight = mass * acceleration due to gravity
17 N = mass of the rifle * 9.81 m/s^2
mass of the rifle = 17 N / 9.81 m/s^2
Now, we can calculate the change in momentum of the rifle:
Change in momentum = mass of the rifle * change in velocity
Assuming that the recoil velocity of the rifle is v, the change in velocity is:
Change in velocity = final velocity - initial velocity
= v - 0
= v
Therefore, the change in momentum of the rifle is:
Change in momentum = (17 N / 9.81 m/s^2) * v
Since momentum is conserved, the change in momentum is equal to the final momentum:
Change in momentum = - Final momentum
= - (0.0036 kg * 231 m/s)
(17 N / 9.81 m/s^2) * v = - (0.0036 kg * 231 m/s)
Solving for v, we get:
v = (- (0.0036 kg * 231 m/s)) / (17 N / 9.81 m/s^2)
v ≈ -0.0115 m/s
Since the recoil of the rifle will be in the opposite direction to the bullet's velocity, the recoil speed of the rifle is approximately 0.0115 m/s.