A stationary mass of 10 kg disintegrates explosively, with the evolution of 4.2*10^5 J,
into three parts, two having equal masses of 3 kg. These two masses move at right
angles to each other. Calculate the speed of the third part.
To solve this problem, we can consider the principle of conservation of momentum and conservation of energy.
Let's assume the initial mass (m_initial) disintegrates into three parts: two masses (m1 and m2) moving at right angles to each other, and the third part of mass m3.
According to the conservation of momentum, the total momentum before the explosion (p_initial) is equal to the total momentum after the explosion (p_final).
p_initial = p_final
To determine the momentum of the third part (p3) after the explosion, we need to calculate the speed of the third part (v3).
Let's break down the given information:
Initial mass (m_initial) = 10 kg
Explosion energy (E) = 4.2 * 10^5 J
Mass of the two parts (m1 and m2) = 3 kg (each)
Mass of the third part (m3) = ?
First, we calculate the momentum before the explosion (p_initial):
p_initial = m_initial * 0 (since the mass is stationary)
= 0 kg m/s
Next, we calculate the momentum after the explosion (p_final). The momenta of the two parts (m1 and m2) can be obtained using the equation p = m * v, where v is the velocity. Since these two parts move at right angles to each other, their momenta are perpendicular to each other and can be added using the Pythagorean theorem.
p1 = m1 * v1
p2 = m2 * v2
The magnitude of the total momentum (p_final) can be calculated as follows:
p_final^2 = p1^2 + p2^2
Now, we can substitute the given values into the equation:
p_final^2 = (3 kg * v1)^2 + (3 kg * v2)^2
Since the magnitudes of the momenta of particles 1 and 2 are equal, we can simplify the equation:
p_final^2 = (3^2 kg^2) * (v1^2 + v2^2)
Next, let's calculate the energy of the explosion (E) using the equation:
E = (1/2) * m_initial * v3^2
Substituting the given values:
4.2 * 10^5 J = (1/2) * 10 kg * v3^2
Now, let's combine the equation for p_final and the equation for E:
p_final^2 = (3^2 kg^2) * (v1^2 + v2^2) = 10^5 kg^2 * v3^2
Since p_final equals p_initial, we have:
10^5 kg^2 * v3^2 = 0
Simplifying the equation:
v3^2 = 0
Taking the square root of both sides, we get:
v3 = 0 m/s
Therefore, the speed of the third part is 0 m/s.