A bullet of mass 6.45 g is fired from a 66.0 cm-long rifle barrel at 237 m/s. The mass of the rifle is 4.50 kg. Determine the acceleration of the bullet as well as the recoil acceleration of the rifle
To find the acceleration of the bullet, we can use Newton's second law of motion, which states that the force acting on an object is equal to its mass multiplied by its acceleration.
1. First, let's convert the mass of the bullet from grams to kilograms:
Mass of the bullet = 6.45 g = 6.45 × 10^(-3) kg
2. Now, let's calculate the force acting on the bullet:
Force = mass × acceleration
Rearranging the formula, we have:
Acceleration = Force / Mass
The force acting on the bullet can be determined using its initial velocity and the length of the barrel. The force is equal to the rate of change of momentum of the bullet:
Force = (mass of the bullet × change in velocity) / time
The change in velocity is the final velocity of the bullet (when it leaves the barrel) minus its initial velocity (when it was at rest in the barrel). Since the bullet was fired, its initial velocity is zero.
Change in velocity = final velocity - initial velocity
= 237 m/s - 0 m/s
= 237 m/s
The time it took for the bullet to travel the length of the barrel can be calculated using the length of the barrel and the speed of the bullet:
Time = Length / Speed
Plugging in the given values:
Time = 66.0 cm / (237 m/s)
= 0.2778 s
Now, we can calculate the force:
Force = (6.45 × 10^(-3) kg × 237 m/s) / 0.2778 s
3. Finally, we can calculate the acceleration:
Acceleration = Force / Mass
Plugging in the values, we get:
Acceleration = [(6.45 × 10^(-3) kg × 237 m/s) / 0.2778 s] / (6.45 × 10^(-3) kg)
= 23697 m/s^2
Thus, the acceleration of the bullet is approximately 23697 m/s^2.
To find the recoil acceleration of the rifle, we'll use the concept of conservation of momentum.
4. The total momentum before the bullet is fired is equal to the total momentum after the bullet is fired. The initial momentum is due to the rifle alone, and the final momentum is due to both the rifle and the bullet. Mathematically, it can be expressed as:
Initial momentum = final momentum
Initial momentum = mass of the rifle × velocity of the rifle before firing (since the rifle was at rest initially)
Final momentum = (mass of the rifle + mass of the bullet) × velocity of the rifle after firing
Equating the two momenta:
mass of the rifle × velocity of the rifle before firing = (mass of the rifle + mass of the bullet) × velocity of the rifle after firing
We can find the velocity of the rifle after firing using the equation:
Velocity of the rifle after firing = (mass of the rifle × velocity of the rifle before firing) / (mass of the rifle + mass of the bullet)
Plugging in the given values:
Velocity of the rifle after firing = (4.50 kg × 0 m/s) / (4.50 kg + 6.45 × 10^(-3) kg)
5. Now, we can calculate the recoil acceleration of the rifle using Newton's second law of motion:
Acceleration = Force / Mass
The force acting on the rifle is equal to its mass multiplied by its acceleration, which can be calculated using the change in velocity and the time taken for the bullet to travel the length of the barrel:
Force = (mass of the rifle × change in velocity) / time
Change in velocity = velocity of the rifle after firing - velocity of the rifle before firing
= [(4.50 kg × 0 m/s) / (4.50 kg + 6.45 × 10^(-3) kg)] - 0 m/s
Plugging in the given values for time and mass of the rifle, we get:
Force = [(4.50 kg × 0 m/s) / (4.50 kg + 6.45 × 10^(-3) kg)] / 0.2778 s
Finally, the recoil acceleration of the rifle can be calculated:
Acceleration = [(4.50 kg × 0 m/s) / (4.50 kg + 6.45 × 10^(-3) kg)] / 0.2778 s) / 4.50 kg
Simplifying the equation above will give you the recoil acceleration of the rifle.
To determine the acceleration of the bullet, we can use Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration:
Force = mass * acceleration
First, we need to convert the mass of the bullet from grams to kilograms:
Mass of bullet = 6.45 g = 6.45 * 10^(-3) kg
Now, we can calculate the force acting on the bullet. The only force acting on the bullet is the force generated by the expanding gases in the rifle barrel, which propels the bullet forward. This force is given by:
Force = mass of bullet * acceleration of bullet
Rearranging the equation, we can solve for the acceleration of the bullet:
Acceleration of the bullet = Force / mass of the bullet
Now, let's calculate the force:
Force = mass of bullet * acceleration of bullet
Force = (6.45 * 10^(-3)) kg * acceleration of bullet
We don't have the value for the force, but we can find it using the principle of conservation of momentum. According to this principle, the momentum of the bullet must be equal to the momentum of the rifle.
The momentum of an object is given by the product of its mass and velocity:
Momentum = mass * velocity
Momentum of bullet = mass of the bullet * velocity of the bullet
Momentum of rifle = mass of the rifle * velocity of the rifle
Since the rifle is initially at rest, the momentum of the rifle is zero:
Momentum of bullet = 0 + momentum of rifle
mass of the bullet * velocity of the bullet = 0 + (mass of the rifle * velocity of the rifle)
Substituting the given values:
(6.45 * 10^(-3)) kg * 237 m/s = 0 + (4.50 kg * velocity of the rifle)
We can solve this equation for the velocity of the rifle:
Velocity of the rifle = (6.45 * 10^(-3)) kg * 237 m/s / 4.50 kg
Velocity of the rifle = 0.03397 m/s (approximately)
Now that we have the velocity of the rifle and the mass of the rifle, we can calculate the force acting on the rifle using the equation:
Force = mass of the rifle * acceleration of the rifle
Rearranging the equation, we can solve for the acceleration of the rifle:
Acceleration of the rifle = Force / mass of the rifle
To find the force acting on the rifle, we can use the equation:
Momenta of bullet + Momenta of rifle = 0
Substituting the known values:
(mass of the bullet * velocity of the bullet) + (mass of the rifle * velocity of the rifle) = 0
Now, we can solve for the force acting on the rifle:
Force = - (mass of the bullet * velocity of the bullet) / (mass of the rifle)
Force = - (6.45 * 10^(-3)) kg * 237 m/s / 4.50 kg
The negative sign indicates that the force exerted on the rifle opposes the forward motion of the bullet.
Finally, we can calculate the acceleration of the rifle:
Acceleration of the rifle = - Force / mass of the rifle
When you substitute the values into the equations, you will find the acceleration of the bullet as well as the recoil acceleration of the rifle.
acceleration time t
0.660 meters = (1/2) a t^2
237 = a t so t = 237/a
then
0.660 = (1/2) a (237^2/a^2)
0.330 a = 237^2
a = 237^2/0.330
momentum is conserved
so
4.50 v rifle = .00645 v bullet at all times
so
a rifle = (.00645/4.50) a bullet
check my arithmetic !