Question

Alex the archaeologist is exploring a cave in the icy mountains of the Himalayas. She is looking for the lost Sankara Stones. The cave she is exploring has a slight incline and is iced over, forcing her to use ice cleats to explore it. After climbing 50 m into the cave, she finds what she is looking for and she takes out her ice pick and hammer to begin the delicate process of removing the stones. Little does she know her hammering has triggered a rock slide 100 m into the cave, releasing a large boulder to slide down the icy incline with no friction. The boulder is released from rest and accelerates down toward her at a constant rate of 0.85 m/s2. She doesn’t notice this until the boulder is only 25 m away from her. Reacting quickly, she pushes herself down the incline towards the cave entrance, knowing the boulder is too large to fit through it. Remembering what she learned in physics, she knows she will accelerate down the inline with the same acceleration as the boulder, thanks to the frictionless surface created by the ice.

(a) What is the boulder’s velocity when Alex first sees it? (Hint: The boulder has moved 25 m from its starting point.)

(b) Now the boulder is 75 m away from the cave entrance and Alex is 50 m away from the entrance. Using your answer to part a, find out how long the boulder has till it reaches the entrance starting from the time Alex sees it.

(c) What minimum initial velocity must Alex give herself so she can escape the cave without being hit by the boulder? (Hint: Use the time you found in part b)

(d) Now since Alex has escaped the cave, she has a new problem. She is sliding towards a cliff that is 10 m away from the cave entrance. The good news is, she is no longer speeding up, but instead slowing down due to friction. Based on your previous answers, how fast is she moving when she escapes the cave?

(e) Using the fact that the static and kinetic coefficients of friction are 0.75 and 0.5, respectively, will she slow down in time and live or will she plummet to her death? Prove your answer mathematically by showing how far she will slide and compare it to the 10 m mentioned above. And no, you do NOT need to know her mass. Assume she is sliding on a horizontal surface.

Answer 1

Vboulder = a t
x boulder = (1/2) a t^2
25 = (1/2) .85 t^2
so
t = 7.67 seconds to do the first 25 m
so at that t
Vboulder = .85 * 7.67 = 6.52 m/s
That is part (a)

now it has to go 75 meters starting with v = 6.52
v = 6.52 + .85 t
where t is time to do those 75 m to entrance
75 = 6.52 t + (1/2) .85 t^2
.425 t^2 + 6.52 t - 75 = 0
t = [ -6.52 +/- sqrt ( 42.5+127.5)]/.85
= 7.67 seconds
That is part (b)

she must do 50 meters in 7.67 seconds with starting speed v and acceleration of .85
50 = v (7.67) + (1/2) .85 (7.67)^2
50 - 25 = 7.67 v
v = 3.26 m/s starting speed