1.A cylinder rolls without slipping on a horizontal plane and then up an incline. The linear speed of the cylinder at the beginning is v. What maximal height h will the cylinder reach?
2.Helicopter of mass M holds stationary in the air fires a missile of mass m<< M with a speed v. Find the recoil speed of the helicopter.
0= Mv + mu
Mv/m = Mu/m
U= -Mv/m
1. To find the maximal height the cylinder will reach, we need to consider the conservation of energy. At the beginning when the cylinder is rolling without slipping on the horizontal plane, it has both kinetic and potential energy. As it moves up the incline, the potential energy increases while the kinetic energy decreases. At the maximal height, all the initial kinetic energy is converted into potential energy.
To find the maximal height, we can equate the initial kinetic energy with the potential energy at the maximal height:
(1/2)mv^2 = mgh
Here,
m is the mass of the cylinder
v is the linear speed of the cylinder at the beginning
g is the acceleration due to gravity
h is the maximal height
Solving the equation for h, we get:
h = (1/2)(v^2) / g
So, the maximal height the cylinder will reach is (1/2)(v^2) / g.
2. To find the recoil speed of the helicopter after firing the missile, we again use the principle of conservation of momentum. Since there are no external forces acting on the system (helicopter + missile), the total momentum before and after firing the missile must be the same.
Before firing the missile, the total momentum is zero because the helicopter is stationary. After firing the missile, the total momentum will still be zero because the momentum of the missile in one direction is balanced by the recoil momentum of the helicopter in the opposite direction.
Using the momentum equation:
(M + m) * 0 = M * v_helicopter_recoil + m * v_missile
Here,
M is the mass of the helicopter
m is the mass of the missile (assumed to be much smaller than M)
v_helicopter_recoil is the recoil speed of the helicopter
v_missile is the speed of the missile
Simplifying the equation, we get:
v_helicopter_recoil = - (m / M) * v_missile
Therefore, the recoil speed of the helicopter is given by the negative of the ratio of the missile mass to the helicopter mass, multiplied by the speed of the missile.