A cannon is rigidly attached to a carriage,which can move along horizontal rails but is connected to a post by a large spring, initially unstretched, of force constant 60000 N/m. The cannon fires a 305 kg projectile at a velocity of 174 m/s directed 38.9◦ above the horizontal.If the mass of the cannon and its carriage is 5537.7 kg, find the recoil velocity of the cannon. Answer in units of m/s
PartB
Determine the maximum extension of the
spring. Answer in units of m
PartC
Find the maximum force the spring exerts on the carriage. Answer in units of N
A. Apply conservation of momentum in the horizontal direction.
B. Apply conservation of energy during comopression
(1/2)kXmax^2 = (1/2)MV^2
V is the velocity from part a.
Solve for Xmax
c) Fmax = k Xmax
Can you help me set up the conservation of momentum in the X direction please? once I figure that out i'm sure I can get it but I can't quite figure that out
The total horizontal momentum of cannon and projectile remains 0
Mp*Vpcos38.9 + Mc*Vrecoil = 0
Mc = cannon mass
Vrecoil = recoil velocity
Vp = projectile mass
Solve for Vrecoil'
V recoil = -(Mp/Mc)*174*cos38.9 m/s
The minus sign means that it recoils backwards, compared to the direction of firing. .
Part A: The recoil velocity of the cannon can be found using the principle of conservation of momentum. Since there are no external horizontal forces acting on the system, the initial horizontal momentum of the system (cannon + carriage) should be equal to the final horizontal momentum after the cannon is fired.
Initial horizontal momentum = Final horizontal momentum
(5537.7 kg)(0 m/s) = (5537.7 kg + 305 kg)(v)
Simplifying the equation:
0 = (5842.7 kg)(v)
v = 0 m/s
Therefore, the recoil velocity of the cannon is 0 m/s.
Part B: The maximum extension of the spring can be determined using Hooke's Law, which states that the force exerted by a spring is proportional to its displacement.
F = kx
Where F is the force exerted by the spring, k is the force constant of the spring, and x is the displacement from its equilibrium position.
In this case, the maximum extension of the spring occurs when it's fully stretched due to the recoil of the cannon. At this point, the force exerted by the spring is equal to the maximum force the spring exerts on the carriage.
F_max = kx_max
The maximum force the spring exerts on the carriage can be calculated using Newton's second law:
F_max = m_carriage * a
Where:
m_carriage = mass of the carriage
a = acceleration of the carriage
The acceleration of the carriage can be determined using Newton's second law:
F_net = m_carriage * a
Where:
F_net = net force on the carriage
The net force on the carriage consists of two components:
1. Horizontal force component due to the recoil of the cannon.
2. Horizontal force component due to the spring.
The horizontal force component due to the recoil of the cannon can be calculated using:
F_recoil = m_projectile * v_projectile
Where:
m_projectile = mass of the projectile
v_projectile = velocity of the projectile
The horizontal force component due to the spring can be calculated using:
F_spring = kx_max
Since the forces are acting in opposite directions, the net force can be found by subtracting the force of the spring from the force of the recoil:
F_net = F_recoil - F_spring
F_net = m_projectile * v_projectile - kx_max
Setting the net force equation equal to the equation for the acceleration:
m_carriage * a = m_projectile * v_projectile - kx_max
Solving for x_max:
x_max = (m_projectile * v_projectile - m_carriage * a) / k
By plugging in the given values:
x_max = (305 kg * 174 m/s - 5537.7 kg * 0 m/s) / 60000 N/m
Simplifying the equation:
x_max = 840.3 m / 60000 N/m
x_max = 0.014 m
Therefore, the maximum extension of the spring is 0.014 m.
Part C: The maximum force the spring exerts on the carriage can be calculated using Hooke's Law:
F_max = kx_max
Plugging in the given values:
F_max = 60000 N/m * 0.014 m
Simplifying the equation:
F_max = 840 N
Therefore, the maximum force the spring exerts on the carriage is 840 N.
Part A: To find the recoil velocity of the cannon, we can use the principle of conservation of momentum. The total momentum before the cannon is fired is equal to the total momentum after the cannon is fired.
The initial momentum before the cannon is fired can be calculated as the sum of the momentum of the projectile and the momentum of the cannon and carriage:
Initial momentum = (mass of projectile) * (velocity of projectile) + (mass of cannon and carriage) * (0)
The final momentum after the cannon is fired can be calculated as the sum of the momentum of the projectile (moving in the opposite direction with recoil velocity) and the momentum of the cannon and carriage:
Final momentum = (mass of projectile) * (-recoil velocity of cannon) + (mass of cannon and carriage) * (velocity of cannon and carriage)
Since the initial momentum equals the final momentum, we can set up the following equation:
(mass of projectile) * (velocity of projectile) = (mass of projectile) * (-recoil velocity of cannon) + (mass of cannon and carriage) * (velocity of cannon and carriage)
Now, we can plug in the given values and solve for the recoil velocity of the cannon:
(305 kg) * (174 m/s) = (305 kg) * (-recoil velocity of cannon) + (5537.7 kg) * (velocity of cannon and carriage)
Solving the equation for the recoil velocity of the cannon will give us the answer in units of m/s.
Part B: To find the maximum extension of the spring, we can use Hooke's law, which states that the force exerted by a spring is directly proportional to the extension or compression of the spring.
The force exerted by the spring can be calculated using the equation:
Force = (spring constant) * (extension or compression of the spring)
In this case, the maximum extension of the spring occurs when the force exerted by the spring is equal to the maximum force it can exert. We can set up the following equation:
(maximum force the spring exerts) = (spring constant) * (maximum extension of the spring)
Now, we can plug in the given values and solve for the maximum extension of the spring. The answer will be in units of meters.
Part C: To find the maximum force the spring exerts on the carriage, we can again use Hooke's law.
The force exerted by the spring can be calculated using the equation:
Force = (spring constant) * (extension or compression of the spring)
In this case, the maximum force the spring exerts on the carriage occurs when the spring is at its maximum extension. We can plug in the maximum extension of the spring (calculated in Part B) into the equation to find the maximum force the spring exerts. The answer will be in units of Newtons.