A Rocket Moves Upward Starting From Rest With An Acceleratio

A Rocket Moves Upward Starting From Rest With an Acceleration Of 2

Calculate the height the rocket reaches after accelerating upward from rest with an acceleration of 29.4 m/s² for 8.00 seconds, then runs out of fuel but continues ascending due to its inertia.

The problem involves analyzing motion under constant acceleration, using kinematic equations to find the final velocity after 8 seconds and the height gained during this period, then considering the continued ascent after fuel exhaustion.

Initial velocity, u = 0 m/s

Acceleration, a = 29.4 m/s²

Time, t = 8.00 s

First, find the velocity at the end of fuel burn:

v = u + a*t = 0 + (29.4)(8.00) = 235.2 m/s

Next, find the height reached during the powered ascent:

s = ut + (1/2)at² = 0 + (1/2)(29.4)(8.00)² = 0 + 0.5 29.4 64 = 941.76 m

The rocket then continues upward after fuel runs out, with initial velocity 235.2 m/s. It will rise until its velocity becomes zero at the peak.

Time to reach the maximum height after fuel exhaustion:

t_peak = v / g = 235.2 / 9.8 ≈ 24.00 s (assuming gravity g = 9.8 m/s²)

The additional height gained during this coasting phase:

s_coast = v t_peak - (1/2) g t_peak² = 235.2 24.00 - 0.5 9.8 * (24.00)²

= 5652.8 - 0.5 9.8 576 = 5652.8 - 2822.4 = 2830.4 m

Finally, total maximum height:

h_total = height during powered ascent + height during coast = 941.76 + 2830.4 ≈ 3772.16 m

Paper For Above instruction

In this problem, we analyze the motion of a rocket that accelerates upward from rest and continues to ascend after the fuel is exhausted. The primary goal is to determine the maximum height reached by the rocket, considering both the powered ascent and the coasting phase after the engines cut off.

Initially, the rocket starts from rest (u=0) with a significant acceleration of 29.4 m/s² over 8 seconds. Applying the basic kinematic equations, the velocity at the end of this period is calculated as v = u + at, which results in 235.2 m/s. Simultaneously, the height it reaches during this powered phase is determined by s = ut + (1/2)at², yielding 941.76 meters.

Once the fuel runs out, the rocket continues to ascend solely under the influence of gravity. The duration of this upward coasting phase is derived from the fact that velocity decreases due to gravity until reaching zero at the maximum height. The time to reach this apex after engine cutoff is v/g = 235.2/9.8 ≈ 24.00 seconds.

The additional height during this coasting period factors in the velocity at engine cutoff and the deceleration due to gravity. Using s = vt - (1/2)gt², the value comes out to approximately 2830.4 meters. Summing this with the initial height during powered ascent, the total maximum height the rocket achieves is roughly 3772.16 meters.

This comprehensive analysis combines the effects of acceleration and gravity, illustrating the application of fundamental kinematic principles to complex motion scenarios in rocketry and physics.

References

  • Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.
  • Tipler, P. A., & Mosca, G. (2008). Physics for Scientists and Engineers. W. H. Freeman.
  • Halliday, D., Resnick, R., & Walker, J. (2014). Fundamentals of Physics. Wiley.
  • University Physics (2014). Mechanics. OpenStax.
  • Hewitt, P. G. (2014). Conceptual Physics. Pearson.
  • Walker, J. (2016). Physics: Principles with Applications. Pearson.
  • Giancoli, D. C. (2013). Physics for Scientists and Engineers. Pearson.
  • Young, H. D., & Freedman, R. A. (2019). University Physics with Modern Physics. Pearson.
  • Reif, F. (2008). Fundamentals of Physics. McGraw-Hill Education.
  • Morin, D. (2008). Introduction to Classical Mechanics. Cambridge University Press.

Related Assignments