D = 10 m. = 30 and vf 8 m/s. Note that the final angle is fixed and cannot be moved, but we are NOT told that the speed should be 8 m/s!

Transcription

1 Mars Probe Your group has been selected to sere on a citizen's panel to ealuate a new proposal to search or lie on Mars. On this unmanned mission, the lander will leae orbit around Mars alling through the atmosphere until it reaches 0,000 meters aboe the surace o the planet. At that time a parachute opens and takes the lander down to 500 meters. Because o the possibility o ery strong winds near the surace, the parachute detaches rom the lander at 500 meters and the lander alls reely through the thin Martian atmosphere with a constant acceleration o 4 m/s² or.0 second. Retrorockets then ire to bring the lander to a sotly to the surace o Mars. A team o biologists has suggested that Martian lie might be ery ragile and decompose quickly in the heat rom the lander. They suggest that any search or lie should begin at least 9 meters rom the center o the lander. This biology team has designed a probe, which is shot rom the lander when it is sitting on the surace o mars by a spring mechanism located at the top center o the lander,.0 meters aboe the Martian surace. To return the data, the probe cannot be more than meters rom the center o the lander. Combining the data acquisition requirements with the biological requirements the team designed the probe to enter the surace o Mars 0 meters rom the center o the base o the lander, at an angle o 30 degrees with the ertical. For the probe to unction properly it must impact the surace with a elocity o at most 8.0 m/s. Can this probe work as designed? I so, what are the speciications o the spring mechanism? I not, is there a design change that will sole the problem?

2 This problem is about projectile motion: y θ 0 V0 h = m x V D = 0 m θ = 30 The inormation and constraints we hae are: h = m D = 0 m θ = 30 8 m/s g = 4 m/s (rom the description o the all o the lander, third sentence rom the top) We do not know the initial angle or speed, and we don t know the inal speed. The initial speed and angle determine the inal speed and angle. Hopeully there is some combination o 0 and θ0 that allows or θ = 30 and 8 m/s. Note that the inal angle is ixed and cannot be moed, but we are NOT told that the speed should be 8 m/s! Since we need the probe to all at some particular spot, let us write the equations or position, or the inal spot: x : D t y : 0 h t gt []

3 Since we don t know anything about the components o the initial elocity, let s relate them to the inal elocity: x gt y or, een better: x gt y [] System [] allows us to rewrite [] in terms o the inal elocity instead o the initial one! D t x 0 h yt gt Or, in terms o angle θ D sin t 0 h cost gt [3] Equations [3] are a system o two equations with two unknowns: t and. Eerything besides these two quantities is known. At this point, we can plug in our numerical alues and sole the system. 0 t 3 0 t t [4] From the irst equation, 0 t [5] Note that ) we are using the angles with respect to the ertical not the usual choice; cos (negatie) because the probe is moing down, ) y

4 We substitute this in the second equation: m/s This is less than 8 m/s! 0 Using [5], we see that this inal speed corresponds to time t.8 s 7. We can now go back to equations [] to ind which initial speed and angle are needed to accomplish this: x y 7. m/s sin m/s gt 7. m/s cos30 4 m/s.8 s 4.8 m/s Or 6.0 m/s 0 0 tan 37 Conclusion: It is possible to launch the probe satisactory with this setup. The probe will enter the ground at 30 degrees as required, with a speed o 7. m/s ( <8 m/s, as desired). The spring mechanism must be set to shoot the probe with initial speed o 6.0 m/s and an angle o 37 degrees with the ertical. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ An alternatie approach Probably the hardest point o the solution aboe is realizing that we need to rewrite the equations in system [] in terms o the inal elocity components, instead o the initial ones.

5 There is a more physically elegant way to approach the problem that inoles less algebraic manipulation o equations. Because graity is constant, the trajectory o a projectile is identical i we time-reerse the motion. In other words, i a probe that starts at A with elocity 0 ends at B with elocity, the same probe launched at B with initial elocity will end at A with inal elocity 0 : y x h = m V0 θ 0 θ = 30 V D = 0 m Note the dierent coordinate system we will use. In this new situation, the equations or the inal position are: x : D sin t y : h cost gt This is already a system with two equations and two unknowns where all quantities are known except or t and! 0 t 3 t t This is exactly the same system as [4].