everyone knows that Batman has a grappling hook installed in the side of the 250

Question

everyone knows that Batman has a grappling hook installed in the side of the 2500 kg, 2 m wide bat mobile. In order to vary the length of the graphing hook, batman has installed a suspension spring at the end of the cable before the hook. The cable is 5 m long and the unstretched length of the 110,000 N/m spring is 2.5 m. We’ll assume the hook is negligibly small. If batman wants to make a 90 degree turn into an alley thats only 3 m wide, centered 13 m from a pole from which the grappling hook will attach, how fast will he have to be going? Give the full range of acceptable speeds (minimum & maximum) in mph that will allow the Batmobile to safely enter the alleyway. (assume frictionless surface)

Explanation / Answer

While taking turn in to the alley the maximum and minimum distance of the batman from the pole are

    Rmax= 12.5 metere and Rmin=11.5 meteres  

Let the corresponding speeds be Vmin and Vmax respectively.

Then

;                 (i)

                    (ii)

Where and are respectively the maximum and minimum extension of the spring. these are respectively xmax= 12.5-7.5= 5 m and 11.5-7.5=4 m

Given k=110000 N/m    (iii)

Substituting these values in equations (i), we get

And

which give

Vmax =52.44 m/sec and Vmin= 45 m/sec

which give

Vmax=52.68 m/sec and Vmin=

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