A space craft made up of two parts with masses m1 = 2.3 kg and m2 = 3.4 kg is held together by explosive bolts against a spring. The spring is compressed by 73.0 cm and has negligible mass. After the bolts are exploded m1 is found to have a speed of 3.6 m/s away from the center of the spacecraft. What is the speed of m2?
To solve this problem, we can apply the principle of conservation of momentum and energy.
First, let's find the total initial mass of the spacecraft, which is the sum of the masses of m1 and m2:
m_total = m1 + m2 = 2.3 kg + 3.4 kg = 5.7 kg
The spacecraft is at rest initially, so the initial total momentum is zero:
p_initial_total = 0 kg·m/s
When the explosive bolts are exploded, the spring pushes the two parts of the spacecraft apart. As a result, the total momentum of the system is conserved, which means the final total momentum is also zero:
p_final_total = 0 kg·m/s
The momentum of an object is calculated using the formula:
momentum = mass × velocity
We are given the mass and velocity of m1:
m1 = 2.3 kg
v1 = 3.6 m/s (away from the center)
Substituting these values into the momentum formula for m1:
p1 = m1 × v1 = 2.3 kg × 3.6 m/s = 8.28 kg·m/s
Since the total final momentum is zero, the momentum of m2 must be equal in magnitude but opposite in direction to that of m1:
p2 = -p1 = -8.28 kg·m/s
Now, let's find the velocity of m2. The momentum of an object is also related to its speed by the formula:
momentum = mass × speed
Rearranging the formula to solve for speed:
speed = momentum / mass
Substituting the values into the speed formula for m2:
speed2 = p2 / m2 = -8.28 kg·m/s / 3.4 kg = -2.435 m/s
Therefore, the speed of m2 is approximately 2.435 m/s in the opposite direction to that of m1.