Scene 3

Analysis Video

Scene

Detailed Analysis

There were a number of large assumptions I had to make to come to my conclusion so I will explain those as I get to them. In order to determine the amount of air resistance that Hunt would feel, I knew I needed to know the helicopter's velocity. However, there was no camera angle in the scene that allowed me to simply use tracker to calculate velocity, so I needed to find a workaround. My starting point was measuring the angle of the rope in the screenshot below (which I assumed to be the farthest back the cargo would go).

I found the angle to be 74.2 degrees. I also was able to calculate the tension of the rope in the y-direction by using the equation Tension = mass x gravity. I found the vertical tension to be 6674.72 newtons. Then, using the law of sines, as shown below, I was able to calculate the tension in the x-direction as well as the rope's total tension.

Now that I knew the tension in the x-direction, I was able to solve for the helicopter's acceleration since Tension = mass x acceleration. I found the acceleration to be 2.766 m/s^2, and the work for that is shown below.

I then made another assumption. This assumption is likely the one that caused the strange answer, however, without being able to calculate the velocity directly in Tracker, this was a necessary step. The only distance I could have possibly used to calculate the velocity of the helicopter was from the center of mass of the hanging mass to the point of attachment of the rope to the helicopter, but only in the x-direction. This is hard to explain so it can be seen better in the work below. The big assumption made here was that the helicopter would be at its full velocity as soon as the rope was at its minimum angle (started at 90 degrees). The reality is that it kept accelerating for a while so my final velocity value was a lot lower than it should have been in reality. I'll get to that in a second. Here is the work to calculate distance.

I first used Tracker to measure the length of the rope (my base distance was Tom Cruz who is 5'8"). The rope is 15.3m long. I then used the law of sines again to calculate the x-component of the distance.

The equation I used to solve for velocity was one of the constant acceleration equations. The equation was:

(final velocity)^2 = (initial velocity)^2 + 2(acceleration)(distance traveled)

Final velocity is what we are solving for, initial velocity is assumed to be 0, acceleration is 2.776 m/s^2, and the distance is 4.17m. I plugged in all the numbers and the velocity I got for the helicopter was 4.81 m/s. To Americanize that for context, that is only 10.76 miles per hour. Clearly, this seems low, so it is safe to assume the helicopter accelerated over a longer distance, we just don't know how much longer. The final step was to calculate the drag force (air resistance) that Ethan Hunt would be feeling.

The equation for drag is as follows:

Drag = 0.5(drag coefficient)(air density)(velocity)^2(surface area)

Typical air density is 1.225 kg/m^3. Velocity is 4.81 m/s. Drag coefficient of a human is about 1.2. The surface area of a 5'8 human is about 1.8 m^2.

This calculates to about 20.406N of drag force. It is difficult to tell how much this affected the outcome. The velocity gets squared for the drag equation so a higher velocity equates to a much higher drag force. It takes 4000N to break a typical human femur, but that is also one of the strongest bones in the body. If we assume the average bone takes about 2000N to break, our 20.4N here seems like it wouldn't even phase a person. This is true. If the force were actually 20.4N, assuming Hunt has the ability to scale the rope without external forces, he should not have much more difficulty here than he normally would. However, if the velocity was actually in the range of 50 m/s, the drag force is now 2205N. This would be near impossible for Hunt to climb.

All in all, the safest guess for this analysis is that this scene is possible, just extremely difficult. The helicopter was definitely flying somewhat slowly due to the cargo it was carrying, it's just a matter of how slow that actually was because velocity is exponential in the drag equation.