Alex is exploring a cave that 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 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)
Any form of guidance or help is greatly appreciated. Thank you.
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
Thank you Thank you Thank you so very much @Damon !! Bless you!
To solve this problem, we can use the equations of motion and apply them to both Alex and the boulder. Let's break it down step by step:
(a) To find the boulder's velocity when Alex first sees it, we need to find its final velocity (v) using the equation of motion:
v^2 = u^2 + 2as
where:
v is the final velocity of the boulder,
u is the initial velocity of the boulder,
a is the acceleration (0.85 m/s^2), and
s is the displacement (25 m).
Since the boulder starts from rest (u = 0), the equation simplifies to:
v = sqrt(2as)
Plugging in the values:
v = sqrt(2 * 0.85 m/s^2 * 25 m)
v ≈ 7.35 m/s
Therefore, the boulder's velocity when Alex first sees it is approximately 7.35 m/s.
(b) To find out how long the boulder has till it reaches the entrance starting from the time Alex sees it, we can use the equation:
s = ut + (1/2)at^2
where:
s is the displacement (75 m),
u is the initial velocity of the boulder (7.35 m/s),
a is the acceleration (0.85 m/s^2), and
t is the time.
Rearranging the equation, we get:
t^2 + (2u/a)t - (2s/a) = 0
Substituting in the values:
t^2 + (2 * 7.35 m/s) / (0.85 m/s^2) * t - (2 * 75 m) / (0.85 m/s^2) = 0
Solving this quadratic equation will give us the time it takes for the boulder to reach the entrance. The positive value of t will be the correct answer.
(c) Once we find the time (t) in part (b) required for the boulder to reach the entrance, we can calculate the minimum initial velocity Alex must give herself to escape the cave without being hit by the boulder.
Since the distance traveled by Alex is the same as the distance traveled by the boulder, we can use the equation:
s = ut + (1/2)at^2
where:
s is the displacement (50 m),
u is the initial velocity of Alex (which we need to find),
a is the acceleration (0.85 m/s^2), and
t is the time found in part (b).
Rearranging the equation, we get:
u = (s - (1/2)at^2) / t
Substituting in the values:
u = (50 m - (1/2) * 0.85 m/s^2 * t^2) / t
Using the value of t found in part (b), we can calculate the minimum initial velocity (u) needed for Alex to escape without being hit by the boulder.