A 14 kg body is moving through space in the positive direction of an x axis with a speed of 220 m/s when, due to an internal explosion, it breaks into three parts. One part, with a mass of 6.4 kg, moves away from the point of explosion with a speed of 190 m/s in the positive y direction. A second part, with a mass of 2.5 kg, moves in the negative x direction with a speed of 460 m/s. What are the (a)x-component and (b)y-component of the velocity of the third part? (c) How much energy is released in the explosion? Ignore effects due to the gravitational force.

The sum of the momenta of the three pieces must equal the original momentum, 14*220 = 3080 kg m/s in the x direction. There is zero iniital momentum in the y direction.

Write equations for the x and y components of total momentum with the component of the third part as the two unknopwns. Solve them.

When done, compare the total KE to the initial KE of the single moving piece (which will be less than final KE), to get the energy release of the explosion

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To solve this problem, we can apply the principle of conservation of momentum and the principle of conservation of kinetic energy.

a) To find the x-component of the velocity of the third part, we need to calculate the x-component of the total momentum before and after the explosion using the principle of conservation of momentum.

Before the explosion:
The total momentum before the explosion is the momentum of the initial body, which can be calculated using the formula:

p_initial = m_initial * v_initial

where
m_initial = 14 kg (mass of the initial body)
v_initial = 220 m/s (initial velocity of the initial body)

p_initial = 14 kg * 220 m/s = 3080 kg·m/s

After the explosion:
The total momentum after the explosion is the vector sum of the individual momenta of the three parts. We need to find the x-component of this total momentum.

For the first part:
p1 = m1 * v1
where
m1 = 6.4 kg (mass of the first part)
v1 = x-component of the velocity of the first part (unknown)

For the second part:
p2 = m2 * v2
where
m2 = 2.5 kg (mass of the second part)
v2 = -460 m/s (negative x-component of the velocity of the second part)

For the third part:
p3 = m3 * v3
where
m3 = mass of the third part (unknown)
v3 = x-component of the velocity of the third part (to be found)

Since the explosion happens in space, the total momentum in the x-direction after the explosion should be zero (conservation of momentum). So we can write:

p_total_x = p1 + p2 + p3 = 0

Substituting the known values:

(6.4 kg * v1) + (2.5 kg * -460 m/s) + (m3 * v3) = 0

Solving this equation will give us the x-component of the velocity of the third part (v3).

b) To find the y-component of the velocity of the third part, we need to calculate the y-component of the total momentum before and after the explosion using the principle of conservation of momentum.

Before the explosion:
The total momentum before the explosion is zero in the y-direction since the body is only moving in the positive x-direction.

After the explosion:
The total momentum after the explosion is the vector sum of the individual momenta of the three parts. We need to find the y-component of this total momentum.

For the first part:
p1_y = m1 * v1_y
where
m1 = 6.4 kg (mass of the first part)
v1_y = 190 m/s (y-component of the velocity of the first part)

For the second part:
p2_y = m2 * v2_y
where
m2 = 2.5 kg (mass of the second part)
v2_y = 0 (no y-component in the velocity of the second part)

For the third part:
p3_y = m3 * v3_y
where
m3 = mass of the third part (unknown)
v3_y = y-component of the velocity of the third part (to be found)

Since the explosion happens in space, the total momentum in the y-direction after the explosion should be zero (conservation of momentum). So we can write:

p_total_y = p1_y + p2_y + p3_y = 0

Substituting the known values:

(6.4 kg * 190 m/s) + (2.5 kg * 0) + (m3 * v3_y) = 0

Solving this equation will give us the y-component of the velocity of the third part (v3_y).

c) To calculate the energy released in the explosion, we can use the principle of conservation of kinetic energy.

Before the explosion:
The kinetic energy before the explosion is given by:

KE_initial = (1/2) * m_initial * (v_initial)^2

After the explosion:
The kinetic energy after the explosion is given by the sum of individual kinetic energies of the three parts:

KE_total = (1/2) * m1 * (v1)^2 + (1/2) * m2 * (v2)^2 + (1/2) * m3 * (v3)^2

The energy released in the explosion can be calculated as the difference between the initial kinetic energy and the total kinetic energy after the explosion:

Energy_released = KE_initial - KE_total

Substituting the known values, we can calculate the energy released in the explosion.