A stick of dynamite (190 g) contains roughly 10 J of chemical energy that is transformed into mechanical energy and heat during an explosion.
A 1000 kg rock in outer space is at rest with respect to a spaceship. An astronaut blows up the rock with one stick of dynamite. Suppose that there are only two pieces of debris after the explosion with masses m_1 = 200 kg and m_2 = 800 kg. What are the speeds of the two pieces with respect to the spaceship? (Assume that we can neglect both heat production during the explosion and the energy needed to break the rock.)
To find the speeds of the two pieces of debris with respect to the spaceship, we can use the principle of conservation of momentum. According to this principle, the total momentum before the explosion is equal to the total momentum after the explosion.
The momentum of an object is given by the product of its mass and velocity, so we can write the equation for conservation of momentum as:
(mass of rock * initial velocity of rock) = (mass of debris 1 * velocity of debris 1) + (mass of debris 2 * velocity of debris 2)
In this case, the initial velocity of the rock is zero as it is at rest with respect to the spaceship. We are given the masses of the debris pieces as m1 = 200 kg and m2 = 800 kg. Let's represent the velocities of the debris pieces as v1 and v2 respectively.
Using the conservation of momentum equation, we can now solve for v1 and v2:
(1000 kg * 0) = (200 kg * v1) + (800 kg * v2)
Simplifying the equation, we get:
0 = 200 kg * v1 + 800 kg * v2
Now, we also know that the chemical energy of the dynamite is transformed into kinetic energy of the debris pieces. The kinetic energy of an object is given by the equation:
Kinetic energy = 1/2 * mass * velocity^2
Using this equation, we can find the kinetic energy of both debris pieces:
Kinetic energy of debris 1 = 1/2 * 200 kg * v1^2
Kinetic energy of debris 2 = 1/2 * 800 kg * v2^2
Since the chemical energy of the dynamite is 10 J, we can set up the following equation:
10 J = Kinetic energy of debris 1 + Kinetic energy of debris 2
Substituting the expressions for the kinetic energies, we get:
10 J = 1/2 * 200 kg * v1^2 + 1/2 * 800 kg * v2^2
Simplifying the equation, we have:
10 J = 100 kg * v1^2 + 400 kg * v2^2
Now, we have two equations:
0 = 200 kg * v1 + 800 kg * v2
10 J = 100 kg * v1^2 + 400 kg * v2^2
Using these two equations, we can solve for v1 and v2, which will give us the speeds of the debris pieces with respect to the spaceship.