Unreasonable Results

(a) What is the initial acceleration of a rocket that has a mass of 1.50×10⁶ kg at takeoff, the engines of which produce a thrust of 2.00×10⁶ N ? Do not neglect gravity.

(b) What is unreasonable about the result? (This result has been unintentionally achieved by several real rockets.)

(c) Which premise is unreasonable, or which premises are inconsistent? (You may find it useful to compare this problem to the rocket problem earlier in this section.)

• The mass of the rocket = 1.50×10⁶ kg.
• The engines produce a thrust = 2.00×10⁶ N.

Apply Newton’s second law of motion as:

$\begin{array}{c}{F}_{net}=ma\\ T-mg=ma\end{array}$

Here, F_net is the net force acting on the system, m is the mass of the rocket, g is the acceleration due to gravity, and a is the acceleration.

Substitute 1.5x10⁶ kg for m, 9.8 m/s² for g, and 2x10⁶ N for T in the above expression, and we get,

$\begin{array}{c}2\times {10}^6\text{\hspace{0.33em}N}-1.5\times {10}^6\text{\hspace{0.33em}k}g\times 9.8{\text{\hspace{0.33em}m/s}}^2=1.5\times {10}^6\text{\hspace{0.33em}k}g\times a\\ 2\times {10}^6\text{\hspace{0.33em}kg}\cdot {\text{m/s}}^2-14.7\times {10}^6\text{\hspace{0.33em}kg}\cdot {\text{m/s}}^2=1.5\times {10}^6\text{\hspace{0.33em}k}g\times a\\ a=\frac{-12.7\times {10}^6\text{\hspace{0.33em}kg}\cdot {\text{m/s}}^2}{1.5\times {10}^6\text{\hspace{0.33em}kg}}\\ a=-8.4667{\text{\hspace{0.33em}m/s}}^2\end{array}$

Hence, the acceleration of the rocket is -8.4667 m/s².

Since the acceleration is less than zero, there is not enough thrust to take off the engine.

The unreasonable premise is that the thrust is not large enough for such mass of the rocket to make it leave the ground.