Problem I - Ethan Hunt needs to jump a gap in the bridge. We know he makes the jump, but the question is, what angle would he need to take off at in order to make the jump?

Known: We know that he does make the jump, and after a quick Google search, we found out his height is 5 feet 7 inches. 

Relevant Quantities: In order to solve for the angle, we need to do trigonometry, but to get there, we need values for the triangle; more specifically, the velocity in the x and y axis at takeoff. To solve for those, the values we need are…

Ø  The size of the gap in the bridge.

Ø  Ethan’s velocity right before he jumps.

Ø  Forces acting on him after takeoff, which since we’re ignoring air resistance, is only gravity.

After analyzing the jump several times in the movie, I was able to come to the conclusion that the gap is roughly three and a half “Tom Cruises” wide (or about 19 feet). In order to find his velocity, I went outside and did several sprints too see what top speed I could achieve. I’m a runner, so I’m in pretty good shape and after several tests I averaged 7 meters per second. Granted his character might be in slightly better shape than I am, we have to consider that he was also wearing pants and a jacket which would slow him down so were going to assume his initial velocity is 7 meters per second. And finally, we know gravity is -9.8 meters per second per second.

Since we know the initial velocity in the X-axis, and we know the distance he has to travel, we can calculate the time it takes to do so, which comes out to 0.83 seconds. We can then use this to figure out the time in the Y-Axis because the times will be identical. We then have to solve for the initial velocity in the Y axis and we do that by taking the 3 known quantities that we have; time,  acceleration due to gravity, and the change in distance in the Y-Axis which is 0 meters. We know the change in distance in the Y-Axis is 0 meters because the side of the bridge that he is landing on is identical to the height of the side that he is taking off on.

Putting those numbers together in the appropriate Kinematic Equation gives us 4.067 meters per second for the initial velocity. Since we now know the initial velocities in both the x and y axis, we can use trigonometry to solve for theta. In the calculator, you would do “the inverse tan function of the value of the side opposite of theta, divided by the value of the side adjacent to theta.

Assuming you entered that properly in the calculator and you’re in degrees mode, not radians, you should have gotten a value for theta being 30.16 degrees. That is the angle that Ethan Hunt has to jump at to just clear the gap.

Problem II – At the end of the film, Ethan Hunt needs to run one mile across town to get to the location where the “Rabbits Foot” is hiding. The question is, what speed does he average on his way there/is that speed even humanly possible?

Known: We know that the “Rabbits Foot” is one mile away because he tells us that in the movie when he is talking on the phone. If you time the run, starting with when he exits the window onto the rooftop, to when he arrives at the door of the other place, it is exactly 100 seconds.

Relevant Quantities: In order to solve for his average velocity on this run, all we need to know is the distance he had to travel, and the time it took him to get there.

Ø  About 1609 meters.

Ø  Exactly 100 seconds.

Knowing that Velocity is equal to the change in distance divided by the change in time, all we have to do is plug in for both those values and we get an average velocity of 16.098 meters per second. That’s about 36 miles per hour.

The highest human “foot” speed ever recorded was 27.79 miles per hour (12.4 meters per second); which was set by Usain Bolt  and recorded during a 100-meter dash in the Olympics. So we can conclude that either Tom Cruise is the fastest man in the world, or this scene is not physically, nor humanly possible!

Problem III – Ethan Hunt leaps off the top of a building running full speed. He is attached to a wire that when engaged, will swing him from one building to another at which point we will detach himself and fall down to the roof of the second building. The question is; is that daring stunt even remotely close to being physically possible?

Known: We know that the height of the first building is 226 meters, and the height of the second building that he needs to land on is only 162 meters.  We assume that Tom Cruise can once again reach his top speed of 7 meters per second when he goes to jump off the building. And finally, we know that the gap between the two buildings is 47.5 meters.  

Relevant Quantities: There are numerous relevant quantities in this problem because there are multiple steps that will go into solving whether it’s possible or not. The relevant quantities include…

Ø  The height of the first building; 226 meters.

Ø  The height of the second building; 162 meters.

Ø  The distance between them; 47.55 meters.

Ø  His initial velocity in the x- axis when he jumps; 7 meters per second.

Ø  Initial velocity in the y-axis when he jumps which we can assume is zero considering he pretty much runs straight right off the top of the building.

Ø  The time that he is in free fall (before the wire is engaged).

Ø  The length of the wire.

Ø  How high off the ground he is when the wire does engage and he starts the swing to the second building.

Ø  Wind resistance during his free fall (after a few calculations it came out to an average of 350 N of force which would reduce his acceleration to -8.1 meters per second per second.

Ø  The time he is physically “swinging” on the pendulum which is roughly 12.6 seconds.

Knowing all of that, you will be able to solve for the distance he travels in the x and y-axis, which comes out to 45.5 meters and -171.1 meters respectively. Using the Pythagorean theorem, we can solve for the length of the rope which comes out to 177.0 meters. Using trig we can solve for the angle of the rope with respect to vertical which comes out to 15 degrees, and we can deduce that at the instant that the wire engages, Ethan Hunt is roughly 54.9 meters above the ground because if he has traveled -171.1 meters vertically, and the building is originally 226 meters tall (basic subtraction).  

At this point, Ethan Hunt is swinging on the end of the wire like a pendulum, and how high he reaches on the other side will depend on his velocity, and the negative work done by air resistance along the way to slow him down.