You are a member of a scientific research team at McMurdo ice station which is funded by the Carver Media Group in Antarctica.

Two members of your research team have recently returned from investigating an incident at a Norwegian research facility. They brought with them a burnt humanoid body with two faces. Since the disturbing discovery several inhabitants of the ice station have disappeared. Frightened, a member of your team decided to flee the station on a fan powered hovercraft but you receive a distress call not long after their escape that their steering and acceleration controls have been jammed and they need your help.

You need to tell the runaway researcher the exact time from your starting time to jump onto your hovercraft as you may only have one shot at this rescue.

Most formulas used in part are are simply kinematic equation. A brief outline of these are found in the notes under Constant Force Motion.

Using the data from the first 20 seconds, we compute the velocity and acceleration of both the runaway ($X,V,A$) and rescue ($x,v,a$) hovercrafts:

Now, to make things as simple as possible, we assume that the rescue cart not only starts from rest, but also accelerates uniformly (constant force from the hovercraft fan) throughout the duration of its rescue mission. Accordingly, the positions in time for both crafts are given by $$X(t)=X_{0{\rm s}}+V t\qquad\mbox{and}\qquad x(t)=x_{0{\rm s}}+\frac{1}{2}at^{2}.$$ Since we want to find the time at which the carts are at the same location, we set the above equations equal and solve for the time: $$t_{\rm jump}=\frac{V\pm\sqrt{V^{2}+2a(X_{0{\rm s}}-x_{0{\rm s}})}}{a}\approx 555.979\,{\rm s}.$$

It then follows, plugging this time back into either position equation, that the location of interest is $$x_{\rm jump}\approx 32359\,{\rm m,}$$ quite a ways away.

Just as you are about to radio the time to jump to the runaway researcher, you realize the steering and acceleration controls have become frozen on your hovercraft and so it continues to accelerate and you cannot change direction. 200m ahead of the point at which you were going to tell the researcher to jump is an ice ravine. At the bottom of the ice ravine, 400m below, is an unfrozen salt water pool surrounded by stalagmites. From the ravine's edge to the pool is 490m and the pool stretches for 900m. You are moving too quickly to survive jumping off the hovercraft, but might survive the fall into the pool by staying on the hovercraft; it has seat belts. You now have a choice to make, to stay on your hovercraft or jump to the runaway researcher's hovercraft. One or both may make it to the pool. Your choice may be the difference between life and death.

To begin, we can calculate the velocities at $x_{\rm jump}$. The runaway craft has a constant velocity ($V=53.64\,{\rm m/s}$), but the rescue craft has velocity given by $v(t)=at.$ Using the time calculated in Part A, we find $$v_{\rm jump}=at_{\rm jump}\approx 116.366\,{\rm m/s}.$$ Furthermore, to find the velocity at the edge of the cliff, we make use of the constant acceleration kinematic equation which does not depend on time ($v_{f}^{2}=v_{i}^{2}+2ad$). Given the distance from the jump to the edge of the cliff, we may immediately find

$$v_{\rm edge}=\sqrt{v_{\rm jump}^{2}+2ad}\approx 116.726\,{\rm m/s},$$

where $d=200\,{\rm m}$.

Now, in order to calculate how far each hovercraft will shoot over the cliff, we need to know how long they will be in the air. Rearranging one of the constant acceleration kinematic equations gives

$$y_{f}=y_{i}+\frac{1}{2}at^{2}\qquad\longrightarrow\qquad t_{\rm air}=\sqrt{\frac{2H}{g}}\approx 9.03\,{\rm s},$$

where $H=400\,{\rm m}$ and $g=9.81\,{\rm m/s^{2}}$.

With this information, we can determine how far each hovercraft lands from the edge of the cliff: $$X_{\rm land}=Vt_{\rm air}\approx 484.395\,{\rm m}\qquad\mbox{and}\qquad x_{\rm land}=v_{\rm edge}t_{\rm air}\approx 1054.09\,{\rm m}.$$ It is clear that we would want to be on the rescue hovercraft.

Surprisingly enough hovercrafts are an expensive piece of kit. Your employer, the Carver Media Group, is concerned by the happenings at the McMurdo ice station and would like you to produce an accident report detailing the events after you lost control of your hovercraft. The accident report should include a detailed computational model that provides the projected motion of the runaway hovercraft.

If python review/ information is needed before attempting this problem, it can be found in the Modeling Motion with VPython notes.

Working first with the runaway craft, we can change the values to reflect those in the problem:

#Parameters and Initial Conditions
runawaycraftv = vector(53.64,0,0)

Next, we can add in the portion of code when the runaway craft is falling ($y>0$), with no ground force:

#Calculation Loop
while runawaycraft.pos.y > 0:
    rate(500)
 
    Fgrav = runawaycraftm*g
    Fnet = Fgrav
 
    runawaycraftp = runawaycraftp + Fnet*dt
    runawaycraft.pos = runawaycraft.pos + (runawaycraftp/runawaycraftm)*dt
 
    runawaycraftMotionMap.update(t, runawaycraftp/runawaycraftm)
 
    t = t + dt

Now for the rescue craft, we copy over all the necessary items and change/add the values/parameters to reflect those of the problem:

#Objects
rescuecraft = sphere(pos=vector(-200,400,0), radius=10, color=color.green)
 
#Parameters and Initial Conditions
rescuecraftm = 1900
rescuecraftv = vector(116.366,0,0)
rescuecraftp = rescuecraftm*rescuecraftv
rescuecrafta = vector(0.2093,0,0)
rescuecraftF = rescuecraftm*rescuecrafta
 
#MotionMap/Graph
rescuecraftMotionMap = MotionMap(rescuecraft, tf, 5, markerScale=1, labelMarkerOrder=False, markerColor=color.orange)
 
#Calculation Loop
while rescuecraft.pos.x < 0:
    rate(500)
 
    Fgrav = rescuecraftm*g
    Fground = -Fgrav
    Fnet = Fgrav + Fground + rescuecraftF
 
    rescuecraftp = rescuecraftp + Fnet*dt
    rescuecraft.pos = rescuecraft.pos + (rescuecraftp/rescuecraftm)*dt
 
    rescuecraftMotionMap.update(t, rescuecraftp/rescuecraftm)
 
    t = t + dt
 
while rescuecraft.pos.y > 0:
    rate(500)
 
    Fgrav = rescuecraftm*g
    Fnet = Fgrav
 
    rescuecraftp = rescuecraftp + Fnet*dt
    rescuecraft.pos = rescuecraft.pos + (rescuecraftp/rescuecraftm)*dt
 
    rescuecraftMotionMap.update(t, rescuecraftp/runawaycraftm)
 
    t = t + dt

#Parameters and Initial Conditions
b = 0.5  #Drag coefficient
 
#Calculation loop
	Fair = -b*dot(...craftp/...craftm,...craftp/...craftm)*...craftp/mag(...craftp)
 
	Fnet = ... + Fair

Solution Code to Part C:

GlowScript 2.9 VPython
 
get_library('https://cdn.rawgit.com/PERLMSU/physutil/master/js/physutil.js')
 
#Window setup
scene.width = 1024
scene.height = 768
scene.center = vector(600,0,0)
 
#Objects
cliff = box(pos=vector(-100,0,0), size=vec(200,800,0), color=color.white)
ravine = box(pos=vector(245,-200, 0), size=vec(490,400,0), color=color.white)
lake = box(pos=vector(940, -200, 0), size=vec(900,400,0), color=color.blue)
runawaycraft = sphere(pos=vector(-200,400,0), radius=10, color=color.red)
rescuecraft = sphere(pos=vector(-200,400,0), radius=10, color=color.green)  # Use runaway as template to add rescue
 
#Parameters and Initial Conditions
g = vector(0,-9.81,0)
b = 0  #Drag coefficient  
 
runawaycraftm = 1500
runawaycraftv = vector(53.64,0,0)    # change velocity to calculated value from part A
runawaycraftp = runawaycraftm*runawaycraftv
 
# Add the next 5 lines for the rescue craft initialed values:
rescuecraftm = 1900
rescuecraftv = vector(116.366,0,0)
rescuecraftp = rescuecraftm*rescuecraftv
rescuecrafta = vector(0.2093,0,0)
rescuecraftF = rescuecraftm*rescuecrafta
 
#Time and time step
t=0
tf=5
dt = 0.01
 
#MotionMap/Graph
runawaycraftMotionMap = MotionMap(runawaycraft, tf, 5, markerScale=1, labelMarkerOrder=False, markerColor=color.orange)
 
# Use runaway motionmap as template for:
rescuecraftMotionMap = MotionMap(rescuecraft, tf, 5, markerScale=1, labelMarkerOrder=False, markerColor=color.orange) 
 
#Calculation Loop
while runawaycraft.pos.x < 0:
    rate(500)
 
    Fgrav = runawaycraftm*g
    Fground = -Fgrav
    Fnet = Fgrav + Fground
 
    runawaycraftp = runawaycraftp + Fnet*dt
    runawaycraft.pos = runawaycraft.pos + (runawaycraftp/runawaycraftm)*dt
 
    runawaycraftMotionMap.update(t, runawaycraftp/runawaycraftm)
 
    t = t + dt
 
# Use given while as tempate for the next 8 lines
# Modify conditional to pos.y and remove ground force
while runawaycraft.pos.y > 0:
    rate(500)
 
    Fgrav = runawaycraftm*g
    Fnet = Fgrav
 
    runawaycraftp = runawaycraftp + Fnet*dt
    runawaycraft.pos = runawaycraft.pos + (runawaycraftp/runawaycraftm)*dt
 
    runawaycraftMotionMap.update(t, runawaycraftp/runawaycraftm)
 
    t = t + dt
 
# Use the 2 working runaway while loops as template to add the following 2 resucue while loops
while rescuecraft.pos.x < 0:
    rate(500)
 
    Fgrav = rescuecraftm*g
    Fground = -Fgrav
    Fnet = Fgrav + Fground + rescuecraftF
 
    rescuecraftp = rescuecraftp + Fnet*dt
    rescuecraft.pos = rescuecraft.pos + (rescuecraftp/rescuecraftm)*dt
 
    rescuecraftMotionMap.update(t, rescuecraftp/rescuecraftm)
 
    t = t + dt
 
while rescuecraft.pos.y > 0:
    rate(500)
 
    Fgrav = rescuecraftm*g
    Fnet = Fgrav   # May include rescuecraftF here too
 
    rescuecraftp = rescuecraftp + Fnet*dt
    rescuecraft.pos = rescuecraft.pos + (rescuecraftp/rescuecraftm)*dt
 
    rescuecraftMotionMap.update(t, rescuecraftp/runawaycraftm)
 
    t = t + dt

Changes made to obtain solution of Part C:

The Carver Media Group is now asking for an accident report for the your hovercraft as well. They want to you simulate the events from the point at which the two hovercrafts meet to when the hovercraft reached the water, however, they want you to model the hovercraft as if it had left the cliff at angle of 27 degrees from the ground. They want this model to be in the same model as the model for the runaway craft.

Solution Code including changes made to obtain the solution for Part C with air drag and Part D adding a 27-degree launch angle:

GlowScript 2.9 VPython
get_library('https://cdn.rawgit.com/PERLMSU/physutil/master/js/physutil.js')
 
#Window setup
scene.width = 1024
scene.height = 768
scene.center = vector(600,0,0)
 
#Objects
cliff = box(pos=vector(-100,0,0), size=vec(200,800,0), color=color.white)
ravine = box(pos=vector(245,-200, 0), size=vec(490,400,0), color=color.white)
lake = box(pos=vector(940, -200, 0), size=vec(900,400,0), color=color.blue)
runawaycraft = sphere(pos=vector(-200,400,0), radius=10, color=color.red)
rescuecraft = sphere(pos=vector(-200,400,0), radius=10, color=color.green)  # Use runaway as template to add rescue
 
#Parameters and Initial Conditions
g = vector(0,-9.81,0)
b = 0.5  #Drag coefficient  change from zero
 
runawaycraftm = 1500
runawaycraftv = vector(53.64,0,0)    # change velocity to calculated value from part A
runawaycraftp = runawaycraftm*runawaycraftv
 
# Add the next 5 lines for the rescue craft initialed values:
rescuecraftm = 1900
rescuecraftv = vector(116.366,0,0)
rescuecraftp = rescuecraftm*rescuecraftv
rescuecrafta = vector(0.2093,0,0)
rescuecraftF = rescuecraftm*rescuecrafta
 
#Add angle of ramp-launch at upper edge of cliff (Extension part D):
thetaindegrees = 27
thetainrad = thetaindegrees*pi/180
dirlaunch = vector(cos(thetainrad),sin(thetainrad),0)
 
#Time and time step
t=0
tf=5
dt = 0.01
 
#MotionMap/Graph
runawaycraftMotionMap = MotionMap(runawaycraft, tf, 5, markerScale=1, labelMarkerOrder=False, markerColor=color.orange)
 
# Use runaway motionmap as template for:
rescuecraftMotionMap = MotionMap(rescuecraft, tf, 5, markerScale=1, labelMarkerOrder=False, markerColor=color.orange) 
 
#Calculation Loop
while runawaycraft.pos.x < 0:
    rate(500)
 
    Fgrav = runawaycraftm*g
    Fground = -Fgrav
    Fdrag = -b*runawaycraftp*mag(runawaycraftp)/(runawaycraftm)**2   #add drag force
    Fnet = Fgrav + Fground + Fdrag   #add drag force
 
    runawaycraftp = runawaycraftp + Fnet*dt
    runawaycraft.pos = runawaycraft.pos + (runawaycraftp/runawaycraftm)*dt
 
    runawaycraftMotionMap.update(t, runawaycraftp/runawaycraftm)
 
    t = t + dt
 
# Add ramp-launch angle at upper-cliff edge:
runawaycraftp = mag(runawaycraftp)*dirlaunch
 
# Use given while as tempate for the next 8 lines
# Modify conditional to pos.y and remove ground force
while runawaycraft.pos.y > 0:
    rate(500)
 
    Fgrav = runawaycraftm*g
    Fdrag = -b*runawaycraftp*mag(runawaycraftp)/(runawaycraftm)**2 #add drag force    
    Fnet = Fgrav + Fdrag         #add drag force
    runawaycraftp = runawaycraftp + Fnet*dt    
    runawaycraft.pos = runawaycraft.pos + (runawaycraftp/runawaycraftm)*dt
 
    runawaycraftMotionMap.update(t, runawaycraftp/runawaycraftm)
 
    t = t + dt
 
# Add ramp-launch angle at upper-cliff edge:
runawaycraftp = mag(runawaycraftp)*dirlaunch
 
 
# Use the 2 working runaway while loops as template to add the following 2 resucue while loops
while rescuecraft.pos.x < 0:
    rate(500)
 
    Fgrav = rescuecraftm*g
    Fground = -Fgrav
    Fdrag = -b*rescuecraftp*mag(rescuecraftp)/(rescuecraftm)**2 #add drag force 
    Fnet = Fgrav + Fground + rescuecraftF + Fdrag
 
    rescuecraftp = rescuecraftp + Fnet*dt
    rescuecraft.pos = rescuecraft.pos + (rescuecraftp/rescuecraftm)*dt
 
    rescuecraftMotionMap.update(t, rescuecraftp/rescuecraftm)
 
    t = t + dt
 
# Add ramp-launch angle at upper-cliff edge:
rescuecraftp = mag(rescuecraftp)*dirlaunch
 
while rescuecraft.pos.y > 0:
    rate(500)
 
    Fgrav = rescuecraftm*g
    Fdrag = -b*rescuecraftp*mag(rescuecraftp)/(rescuecraftm)**2 #add drag force 
    Fnet = Fgrav + Fdrag   #add drag force  
 
    rescuecraftp = rescuecraftp + Fnet*dt
    rescuecraft.pos = rescuecraft.pos + (rescuecraftp/rescuecraftm)*dt
 
    rescuecraftMotionMap.update(t, rescuecraftp/runawaycraftm)
 
    t = t + dt

Changes made to the solution for Part C, adding air drag (boxed in red) and Part D adding a 27-degree launch angle (boxed in purple):