Answer:
a. To find the speed of the larger piece immediately after the explosion, we can use the principle of conservation of momentum.
The initial momentum of the system is given by the mass of the object (2.0 kg) multiplied by its initial velocity (6.0 m/s). This is equal to the sum of the momenta of the two pieces after the explosion.
Let's denote the velocity of the larger piece as v1 and the velocity of the smaller piece as v2. Since the smaller piece stops relative to the ice, its final velocity is 0 m/s.
Using the conservation of momentum, we can write:
(initial momentum) = (momentum of larger piece) + (momentum of smaller piece)
(2.0 kg)(6.0 m/s) = (1.5 kg)(v1) + (0.5 kg)(0 m/s)
Simplifying the equation, we have:
12.0 kg·m/s = 1.5 kg·m/s + 0 kg·m/s
Now, we can solve for v1:
1.5 kg·m/s = v1
So, the speed of the larger piece immediately after the explosion is 1.5 m/s.
b. To find the gain in kinetic energy of the system as a result of the explosion, we need to calculate the total kinetic energy before and after the explosion.
The initial kinetic energy of the system is given by the equation:
KE_initial = (1/2)mv^2
Substituting the values, we have:
KE_initial = (1/2)(2.0 kg)(6.0 m/s)^2
Simplifying the equation, we get:
KE_initial = 36.0 J
After the explosion, the smaller piece stops, so its kinetic energy becomes zero. The kinetic energy of the larger piece can be calculated using the equation:
KE_final = (1/2)mv^2
Substituting the values, we have:
KE_final = (1/2)(1.5 kg)(1.5 m/s)^2
Simplifying the equation, we get:
KE_final = 1.69 J
The gain in kinetic energy of the system as a result of the explosion can be calculated by subtracting the initial kinetic energy from the final kinetic energy:
Gain in kinetic energy = KE_final - KE_initial
Gain in kinetic energy = 1.69 J - 36.0 J
Gain in kinetic energy = -34.31 J
Therefore, the gain in kinetic energy of the system as a result of the explosion is -34.31 J.
