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.
Before the explosion, both the rock and the dynamite stick are at rest with respect to the spaceship, so their momentum is zero.
After the explosion, the total mass of the debris is the sum of the masses of the two pieces: m_total = m1 + m2 = 200 kg + 800 kg = 1000 kg.
Let's assume v1 is the velocity of debris piece 1 and v2 is the velocity of debris piece 2.
Using the conservation of momentum, we have:
0 = m1 * v1 + m2 * v2
To find the velocities, we also need to take into account the energy released during the explosion. We are given that the dynamite stick contains 10 J of chemical energy. This energy is transformed into mechanical energy of motion of the debris.
The total energy released during the explosion is given by:
E = (1/2) * m_total * (v1^2 + v2^2)
Substituting the values, we have:
10 J = (1/2) * 1000 kg * (v1^2 + v2^2)
Simplifying, we get:
10 J = 500 kg * (v1^2 + v2^2)
Dividing both sides by 500 kg, we obtain:
0.02 J/kg = v1^2 + v2^2
Now, we have two unknowns, v1 and v2. However, we can make one more assumption: the explosion releases the energy equally between the two debris pieces. This means v1 should be the same as v2.
So, we can rewrite the equation as:
0.02 J/kg = 2 * v1^2
Simplifying further:
0.01 J/kg = v1^2
To find the value of v1, we take the square root:
v1 = sqrt(0.01 J/kg)
v1 = 0.1 m/s
Since v1 = v2, the speed of both debris pieces with respect to the spaceship is 0.1 m/s.