A lunar lander must adjust its horizontal velocity by to the left (define left as the…

PHYSICS · Multi-part · CBSE · Physics · Force and Laws of Motion · Force and Laws of Motion

Problem

250\,\text{kg}

lunar lander must adjust its horizontal velocity by

\Delta v = 12\,\text{m\,s}^{-1}

to the left (define left as the positive direction) within a

5.0\,\text{s}

maneuvering window. Engineers propose using a single side-thruster that produces a force which starts at zero, rises uniformly to a maximum value

F_{\text{max}}

at time

t = \tau

, and then immediately shuts off (so the force–time graph is a right-triangle pulse of base

\tau

and height

F_{\text{max}}

). Neglect all other external horizontal forces. (a) Derive an expression that relates the required impulse to

F_{\text{max}}

and

\tau

for the triangular force pulse. (b) Using the

5.0\,\text{s}

maneuvering window, determine the minimum value of

F_{\text{max}}

that achieves the required change in velocity without exceeding the structural limit of

1600\,\text{N}

. State the corresponding value of

\tau

. (c) If the structural limit is lowered to

1000\,\text{N}

, determine quantitatively whether the triangular pulse alone can still achieve the required velocity change within

5.0\,\text{s}

. Justify your answer. (d) Propose an alternative piecewise force profile that obeys the

1000\,\text{N}

limit yet delivers the required impulse within

5.0\,\text{s}

. Describe the profile and derive the necessary parameters so that the goal is met.