Question

A 40kg body is moving in the direction of the positive x-axis with a speed of 373 m/s when, an internal explosion occurs and it breaks into three pieces. One part, whose mass is 5.00kg, moves away from the point of explosion with a speed of 391 m/s along the positive y-axis. A second fragment, whose mass is 3.50 kg, moves away from the point of explosion with a speed of 429 m/s along the negative x-axis. How much energy was released in the explosion? Give your answer in megajoules (MJ)?

Answer 1

The energy released in the explosion is approximately 1.77819 megajoules.

Given values:

`\[ m_{\text{total}} = 40 \, \text{kg} \]`

`\[ v_{\text{total}} = 373 \, \text{m/s} \]`

Calculating total kinetic energy before the explosion:

`\[ KE_{\text{before}} = (1)/(2) * 40 \, \text{kg} * (373 \, \text{m/s})^2 \]`

`\[ KE_{\text{before}} = 2.7817 * 10^6 \, \text{Joules} \]`

Now, let's compute the kinetic energy of each fragment after the explosion:

Kinetic energy of Fragment 1:

`\[ KE_1 = (1)/(2) * 5.00 \, \text{kg} * (391 \, \text{m/s})^2 \]`

`\[ KE_1 = 382547.5 \, \text{Joules} \]`

Kinetic energy of Fragment 2:

`\[ KE_2 = (1)/(2) * 3.50 \, \text{kg} * (429 \, \text{m/s})^2 \]`

`\[ KE_2 = 643262.5 \, \text{Joules} \]`

Total kinetic energy after the explosion:

`\[ KE_{\text{after}} = KE_1 + KE_2 \]`

`\[ KE_{\text{after}} = 382547.5 \, \text{Joules} + 643262.5 \, \text{Joules} \]`

`\[ KE_{\text{after}} = 1026810 \, \text{Joules} \]`

Energy released in the explosion:

`\[ \text{Energy released} = KE_{\text{before}} - KE_{\text{after}} \]`

`\[ \text{Energy released} = 2.7817 * 10^6 \, \text{Joules} - 1026810 \, \text{Joules} \]`

`\[ \text{Energy released} = 1.77819 * 10^6 \, \text{Joules} \]`

Now, let's convert the energy released to megajoules (MJ):

`\[ \text{Energy released in MJ} = \frac{1.77819 * 10^6 \, \text{Joules}}{10^6 \, \text{Joules/MJ}} \]`

`\[ \text{Energy released in MJ} = 1.77819 \, \text{MJ} \]`

Answer 2

The energy that was released during the explosion is 2,282,771.1 J.

How to calculate the change in energy?

The energy that was released during the explosion is calculated by applying the following formula.

The total initial kinetic energy is;

K.Ei = ¹/₂mv²

where;

• m is the mass of the body
• v is the speed of the body

K.Ei = ¹/₂ x 40 x (373²)

K.Ei = 2,782,580 J

The total final kinetic energy is;

K.Ey = ¹/₂ x 5 x (391²)

K.Ey = 382,202.5 J

K.Ex = ¹/₂ x 3.5 x (429²)

K.Ex = 322,071.75 J

The resultant final velocity is;

K.Ef = √ (382,202.5² + 322,071.75 )

K.Ef = 499,808.93 J

The lost in energy is;

ΔK,E = 499,808.93 J - 2,782,580 J

ΔK,E = -2,282,771.1 J