A missile launcher, mass M, at rest on a horizontal track fires a missile, mass m, horizontally.
a) If the launch speed of the missile is v o , what (effective) coefficient of (kinetic) friction is
required in order to stop the launcher within x meters? (b) Missile launcher mass is 4400 kg,
missile mass is 110 kg, missile launch speed is 450 m/s, the effective coefficient of friction on
the launcher is 0.75. Find x.
(a)
Law of conservation of lineat momentum
0=mvₒ-Mu
u= mvₒ/M
Law of conservation of energy
M•u²/2=μ•Mg•x
μ = u²/2• g•x = (mvₒ)²/2•M²• g•x
(b) x = (mvₒ)²/2•M²• g• μ
To answer part (a) of the question, we will use the principle of conservation of momentum.
The momentum of the system before the launch is zero since both the launcher and the missile are at rest. The momentum after the launch will also be zero since the launcher is required to stop within x meters.
According to the principle of conservation of momentum, the total momentum before the launch is equal to the total momentum after the launch. The momentum of the missile is given by the equation:
p_m = m * v_o
where p_m is the momentum of the missile, m is the mass of the missile, and v_o is the launch speed of the missile.
Since the momentum of the launcher is initially zero, it will acquire an equal but opposite momentum (p_m) after the launch in order to have a total momentum of zero.
The friction force acting on the launcher can be calculated using Newton's second law, which states that the net force on an object is equal to the product of the mass of the object and its acceleration. In this case, the net force acting on the launcher is the force of friction.
The acceleration of the launcher can be determined using the equation:
F_friction = m * a
where F_friction is the force of friction and m is the mass of the launcher.
The force of friction can be calculated by multiplying the coefficient of kinetic friction (μ) with the normal force (N), which is the weight of the launcher:
F_friction = μ * N
The normal force can be determined using the equation:
N = M * g
where M is the mass of the launcher and g is the acceleration due to gravity.
Now we can substitute the above equations to find the effective coefficient of kinetic friction required to stop the launcher within x meters.
For part (b), the given values are:
M (mass of launcher) = 4400 kg
m (mass of missile) = 110 kg
v_o (launch speed of missile) = 450 m/s
μ (coefficient of friction) = 0.75
To find x, we will use the equation of motion for the launcher:
x = (v_o^2) / (2 * a)
where a is the acceleration of the launcher.
To calculate a, we need to determine the friction force acting on the launcher:
F_friction = μ * N
Since the normal force N is equal to the weight of the launcher, we can calculate it using:
N = M * g
Now, using the equations above, we can find the effective coefficient of kinetic friction and x.