A gun of mass M fires a shell of mass m and recoils horizontally. If the barrel of the gun is horizontal and the shell travels along it with speed v, find the: (a) speed with which the barrel begins to recoil. (b) constant force required to bring the gun to rest in 3 seconds.
(a) To find the speed with which the barrel begins to recoil, we can use the principle of conservation of momentum. According to this principle, the total momentum before the firing of the gun is equal to the total momentum after the firing.
Before firing:
The gun and the shell together have a total initial momentum of 0, as they are at rest.
After firing:
The gun recoils with a final velocity v_g, and the shell moves with a velocity v. The mass of the gun is M, and the mass of the shell is m.
The momentum of the gun after firing is given by: p_g = M * v_g.
The momentum of the shell after firing is given by: p_s = m * v.
According to the conservation of momentum, the total momentum after firing must be equal to 0 (since the total momentum before firing is 0).
Therefore, we have:
p_g + p_s = 0
M * v_g + m * v = 0
Rearranging the equation, we can solve for the velocity of the gun:
M * v_g = -m * v
v_g = -m * v / M
So, the speed with which the barrel begins to recoil is given by:
v_g = -m * v / M
(b) To find the constant force required to bring the gun to rest in 3 seconds, we can use Newton's second law of motion.
The net force acting on an object is equal to the rate of change of its momentum. In this case, the gun is being brought to rest, so its final momentum is 0. The initial momentum of the gun is M * v_g (which we found in part (a)).
The change in momentum is given by:
change in momentum = final momentum - initial momentum
0 - (M * v_g) = -(M * v_g)
The time taken to bring the gun to rest is 3 seconds. Therefore, the constant force required to bring the gun to rest in 3 seconds is given by:
force = change in momentum / time
force = -(M * v_g) / t
force = -(M * (-m * v / M)) / t
force = m * v / t
So, the constant force required to bring the gun to rest in 3 seconds is given by:
force = m * v / t
To solve this problem, we can use the principle of conservation of linear momentum.
According to this principle, the total momentum before the firing of the shell is equal to the total momentum after the firing.
Let's denote the initial velocity of the gun as V_gun (which is zero), and the final velocity of the gun as V'_gun. The initial velocity of the shell is v, and the final velocity of the shell is zero (as it comes to rest).
So, the total initial momentum is given by:
P_initial = m * v + M * V_gun
And the total final momentum is given by:
P_final = m * 0 + M * V'_gun
Since the total momentum is conserved, we have:
P_initial = P_final
m * v + M * V_gun = M * V'_gun
Now, let's solve part (a):
(a) To find the speed with which the barrel begins to recoil, we need to determine the final velocity of the gun (V'_gun). Since the gun initially had zero velocity, the final velocity of the gun will be in the opposite direction to the velocity of the shell.
Therefore, V'_gun = -(m * v) / M
Now, let's solve part (b):
(b) To find the constant force required to bring the gun to rest in 3 seconds, we can use Newton's second law:
F = m * a
where F is the force, m is the mass, and a is the acceleration.
Since the gun is to be brought to rest, the final velocity of the gun is zero, and the initial velocity is V_gun = 0.
Using the equation of motion:
V_final = V_initial + a * t
where V_final is the final velocity, V_initial is the initial velocity, a is the acceleration, and t is the time.
Substituting the values:
0 = 0 + a * 3
This implies:
a = 0
Thus, the constant force required to bring the gun to rest in 3 seconds is zero.
In summary:
(a) The speed with which the barrel begins to recoil is -(m * v) / M.
(b) The constant force required to bring the gun to rest in 3 seconds is zero.