Answer:
a) v = 786.93 m/s
b) v = 122.40 m/s
Explanation:
a) To find the average exhaust speed (v) of the engine we can use the following equation:
[tex] F = \frac{v\Delta m}{\Delta t} [/tex]
Where:
F: is the thrust by the engine = 5.26 N
Δm: is the mass of the fuel = 12.7 g
Δt: is the time of the burning of fuel = 1.90 s
[tex]v = \frac{F*\Delta t}{\Delta m} = \frac{5.26 N*1.90 s}{12.7 \cdot 10^{-3} kg} = 786.93 m/s[/tex]
b) To calculate the final velocity of the rocket we need to find the acceleration.
The acceleration (a) can be calculated as follows:
[tex] a = \frac{F}{m} [/tex]
In the above equation, m is an average between the mass of the engine plus the rocket case mass and the mass of the engine plus the rocket case minus the fuel mass:
[tex]m = \frac{(m_{engine} + m_{rocket}) + (m_{engine} + m_{rocket} - m_{fuel})}{2} = \frac{2*m_{engine} + 2*m_{rocket} - m_{fuel}}{2} = \frac{2*25.0 g + 2*63.0 g - 12.7 g}{2} = 81.65 g[/tex]
Now, the acceleration is:
[tex] a = \frac{5.26 N}{81.65 \cdot 10^{-3} kg} = 64.42 m*s^{-2} [/tex]
Finally, the final velocity of the rocket can be calculated using the following kinematic equation:
[tex]v_{f} = v_{0} + at = 0 + 64.42 m*s^{-2}*1.90 s = 122.40 m/s[/tex]
I hope it helps you!
(a) The average exhaust speed of the engine is 400 m/s.
(b) The final velocity of the rocket case is 158.63 m/s.
The given parameters;
The average exhaust speed of the engine is calculated as follows;
[tex]F = ma\\\\F = m\frac{\Delta v}{t} \\\\\Delta v = \frac{Ft}{m} \\\\\Delta v = \frac{5.26 \times 1.9}{0.025} \\\\\Delta v = 400 \ m/s[/tex]
The impulse received by the rocket case when the engine exerts the given force at the given time;
[tex]m\Delta V = Ft\\\\m\Delta V = 5.26 \times 1.9\\\\m\Delta V = 9.994 \ kgm/s[/tex]
The final velocity of the rocket case is calculated as follows;
[tex]m(v_f - v_0) = 9.994\\\\0.063(v_f -0) = 9.994\\\\v_f = \frac{9.994}{0.063} \\\\v_f = 158.63 \ m/s[/tex]
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