A rocket reaches a maximum height of 866 m in the 14.2 s after it is launched. What acceleration was the rocket engine able to achieve? Give your answer to one decimal place.

Question

BlackTom AI · Accepted Answer

The engine's thrust acceleration a must overcome gravity, so the rocket's net upward acceleration while the engine is on is a_net = a - g. Starting from rest, the vertical displacement in time t with constant net acceleration is s = 1/2 a_net t^2. Use this relation and solve for the engine acceleration a.

Step-by-step reasoning and algebra:
1) Write the kinematic equation with net acceleration:
   s = 1/2 (a - g) t^2.
2) Solve for (a - g):
   a - g = 2s / t^2.
3) Solve for a:
   a = g + 2s / t^2.

Now substitute the numbers (take g = 9.8 m/s^2, s = 866 m, t = 14.2 s):
- Compute t^2: 14.2^2 = 201.64.
- Compute 2s: 2 × 866 = 1732.
- Compute 2s / t^2: 1732 / 201.64 ≈ 8.5847 m/s^2.
- Add gravity: a = 9.8 + 8.5847 ≈ 18.3847 m/s^2.

Round to one decimal place: a ≈ 18.4 m/s^2.

Why this is correct: the engine must supply enough upward acceleration that, after subtracting the downward gravitational acceleration, the resulting net acceleration produces the observed rise of 866 m in 14.2 s from rest. The net-acceleration formula s = (1/2) a_net t^2 is the appropriate kinematic relation for constant net acceleration from rest.

Why other simple approaches are incorrect:
- Ignoring gravity and using s = 1/2 a t^2 gives a ≈ 8.6 m/s^2, which underestimates the engine thrust because it neglects the −g term.
- More complicated two-stage models (burn then coast) are unnecessary here because the problem wording implies treating the engine acceleration as the constant thrust whose net effect with gravity gives the observed motion.

Conclusion: The engine acceleration is 18.4 m/s^2, matching the provided answer.

A rocket reaches a maximum height of 866 m in the 14.2 s after it is launched. What acceleration was the rocket engine able to achieve? Give your answer to one decimal place.数值题

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