A bullet of mass 15g is short from a gun 15kg with a muzzle velocity of 200m/s if the bullet is 20cm calculate;(a)the acceleration of the bullet
(b)the recoil velocity of the gun
what does the length of the bullet matter?
To calculate the acceleration of the bullet, we can use Newton's second law of motion, which states that the force exerted on an object is equal to its mass times its acceleration (F = m * a).
Given that the bullet has a mass of 15 grams (0.015 kg), the velocity of the bullet is 200 m/s, and the bullet is shot from a gun with a mass of 15 kg, we can start by finding the force exerted on the bullet.
(a) The force exerted on the bullet can be calculated using the formula:
Force (F) = mass (m) * acceleration (a)
The mass of the bullet, m = 0.015 kg, and we need to find the acceleration, so we rearrange the formula to solve for acceleration:
a = F / m
To find the force exerted on the bullet, we can use the equation of motion:
Force (F) = mass of the gun (M) * change in velocity (Δv)
The mass of the gun, M = 15 kg, and the change in velocity is the muzzle velocity of the bullet, which is 200 m/s. So, we have:
F = M * Δv
= 15 kg * 200 m/s
= 3000 kg m/s
Now, we can substitute this force value back into the first formula to find the acceleration of the bullet:
a = F / m
= 3000 kg m/s / 0.015 kg
= 200,000 m/s²
Therefore, the acceleration of the bullet is 200,000 m/s².
(b) To calculate the recoil velocity of the gun, we can use the concept of conservation of momentum. According to this principle, the momentum before an event is equal to the momentum after the event.
The initial momentum of the system (the bullet and the gun) is zero since they are initially at rest. The final momentum is zero as well because the bullet is shot in one direction, and the gun recoils in the opposite direction.
Let's denote the recoil velocity of the gun as V.
Then, the momentum before the event is equal to the momentum after the event:
Initial momentum = Final momentum
0 = (mass of the bullet * velocity of the bullet) + (mass of the gun * recoil velocity of the gun)
Substituting the given values, we have:
0 = (0.015 kg * 200 m/s) + (15 kg * V)
Simplifying the equation:
0 = 3 kg m/s + 15 kg * V
Now, solve for V by isolating the variable:
15 kg * V = -3 kg m/s
V = -3 kg m/s / 15 kg
V = -0.2 m/s
However, the negative sign indicates the direction, which means the gun moves in the opposite direction of the bullet. So the recoil velocity of the gun is 0.2 m/s in the opposite direction.
Therefore, the recoil velocity of the gun is 0.2 m/s opposite to the direction of the bullet.
To calculate the acceleration of the bullet and the recoil velocity of the gun, we can use the principle of conservation of momentum.
According to this principle, the total momentum before the bullet is fired is equal to the total momentum after the bullet is fired. The momentum of an object is given by the product of its mass and velocity.
(a) The acceleration of the bullet:
To find the acceleration of the bullet, we can use the formula for acceleration:
acceleration = change in velocity / time
Since we are given the muzzle velocity of the bullet (200 m/s) and the length of the barrel (20 cm), we need to convert the length into time.
The time it takes for the bullet to travel the length of the barrel can be calculated using the formula:
time = distance / velocity
Given:
Distance = 20 cm = 0.2 m
Velocity = 200 m/s
Substituting these values into the formula, we get:
time = 0.2 m / 200 m/s = 0.001 s
Now, we can use the formula for acceleration:
acceleration = change in velocity / time
Before firing, the bullet was at rest, so the initial velocity is 0 m/s. The final velocity is the muzzle velocity of the bullet, which is 200 m/s. Therefore, the change in velocity is:
change in velocity = final velocity - initial velocity
= 200 m/s - 0 m/s
= 200 m/s
Substituting the values into the formula, we get:
acceleration = (200 m/s) / (0.001 s)
= 200,000 m/s^2
Therefore, the acceleration of the bullet is 200,000 m/s^2.
(b) The recoil velocity of the gun:
To find the recoil velocity of the gun, we can use the principle of conservation of momentum. The total momentum before the bullet is fired is equal to the total momentum after the bullet is fired.
Before the bullet is fired, the gun and the bullet are both at rest, so the total initial momentum is 0.
The total final momentum is given by the product of the bullet's mass and velocity, which is:
momentum = mass * velocity
Given:
Mass of the bullet = 15 g = 0.015 kg
Muzzle velocity of the bullet = 200 m/s
Substituting these values into the formula, we get:
momentum = (0.015 kg) * (200 m/s)
= 3 kg m/s
According to the conservation of momentum, the total initial momentum is equal to the total final momentum.
Total initial momentum = Total final momentum
0 = mass of the gun * recoil velocity
Rearranging the equation, we find:
recoil velocity = - (0) / mass of the gun
= 0 m/s
Therefore, the recoil velocity of the gun is 0 m/s.