Answer:
F = 200 [N]
Explanation:
To solve this problem we must use the principle of conservation of linear momentum, which can be calculated by means of the following equation.
[tex]P=m*v[/tex]
where:
P = lineal momentum [kg*m/s]
m = mass [kg]
v = velocity [m/s]
Now we must understand that the momentum is conserved before and after the firing of the cannon. Before firing the cannon we have the mass of the cannon and mass of the cannonball together at rest (speed = 0). After firing the cannon the cannonball moves forward with positive speed, while the cannon moves back (negative), in this way knowing the masses of each one we can determine the speed of the cannon.
[tex](m_{cannon}+m_{ball})*v_{1}=-(m_{cannon}*v_{2})+(m_{ball}*v_{3})[/tex]
where:
m_cannon = 2500 [kg]
m_ball = 2.5 [kg]
v₁ = 0 (velocity of the group before firing)
v₂ = velocity of the cannon after firing [m/s]
v₃ = 160 [m/s] (velocity of the cannonball after firing)
[tex](2500+2.5)*0 = -(2500*v_{2})+(2.5*160)\\v_{2}=0.16[m/s][/tex]
Now using the following equation of kinematics, we can calculate the acceleration.
[tex]v_{f}=v_{o}-a*t[/tex]
where:
Vf = final velocity = 0 (cannon comes to rest)
Vo = initial velocity = 0.16 [m/s]
a = acceleration [m/s²]
t = time = 2 [s]
[tex]0 = 0.16 - a*2\\2*a=0.16\\a = 0.08 [m/s^{2}][/tex]
With the value of acceleration, we can use Newton's second law which tells us that the forces acting on a body is equal to the product of mass by acceleration.
ΣF = m*a
where:
F = force [N] (units of Newtons)
m = mass = 2500 [kg]
a = acceleration = 0.08 [m/s²]
[tex]F = 2500*0.08\\F = 200 [N][/tex]
