# How High Is This Cliff, and Why Would You Jump Off It?

The internet has always been a great source of physics problems. Check out this video. It’s sort of crazy. (Hat tip to @fnoschese and @maalmeida85 on Twitter)

So what kinds of questions can you ask with this video? How about the following?

- What would make some think this is a good idea?
- How high is the cliff?
- How far horizontally did he travel during the jump?
- What is his speed when he impacts the water?
- Estimate how deep into the water he went along with the average stopping acceleration (these are connected).
- What about that scene from Captain America: The Winter Soldier where Cap did something like this from a plane? How does this jump compare?
- How much influence does the air drag have on his final speed? Is it safe to just ignore it?
- Can you measure the speed of sound from this video? (I bet you didn’t see that one coming).

OK, let’s get started. I’m probably not going to answer all these questions — but you can finish whatever is leftover.

**How fast is he going when he hits the water?**

This isn’t the best question from above, but you sort of need it to find out the height of the cliff. It’s still a cool question though since it really only needs the fall time from the video. Even for a simple measurement like this, I like to use Tracker Video Analysis (it’s free and awesome).

By marking the starting and ending frame of the fall, I get a time interval of 2.933 seconds. That’s quite a long time to contemplate your life choices on the way down. But with this time and the acceleration of a free falling object, I can determine the drop height.

I’m going to do this starting from fundamental ideas — the first thing is the definition of acceleration in 1 dimension:

Here I am using v1 for the initial vertical velocity and I’ll assume (for now) that it has a value of 0 m/s. For an object that is falling ONLY under the influence of the gravitational force, it will have a vertical acceleration of -g where g = 9.8 m/s². So, from this expression I can go ahead and find the final velocity (which is another question — I guess I’m going out of order).

Using my value for the time interval, I get a final velocity of -28.74 m/s (the negative means he is moving down — which is obvious). If you don’t have an intuitive feeling for the magnitude of this speed, I’ll convert it into miles per hour. He impacts at 64.3 mph. Damn.

**How high is the cliff?**

This is what you really want to know, right? I already know the starting and ending velocity of the crazy jumper, so with that I just need the definition of the average velocity (in one dimension).

Yes, there are two definitions for average velocity. The first definition is the rate of change of position. The other definition is actually an average (final plus initial divided by 2). Note: the second version only works if the acceleration is constant. But still, I know the velocities and the time so that I can solve for the change in position (also known as the height).

Of course you could just plop in the values for the final velocity and time to get the drop height — but I don’t want to do that. Instead, I am going to put in the expression for the final velocity into this equation (the final velocity from before).

Yup, this is basically one of the kinematic equations (for the special case of a zero starting velocity). Now I only need to put in my value for the time interval. With that, I get a change in position of -42.15 meters (138.3 feet for Imperial people). Here the negative sign just means he moved down — which of course we already knew.

**What is the speed of sound?**

OK, not what is the ACTUAL speed of sound — but rather: can I measure the speed of sound from this video? So, check it out. The guy hits the water and you can see the splash. The visual of the splash travels from the water to the camera at the speed of light (3 x 10⁸ m/s — or really darn fast). This is basically instantaneous.

Next, the sound of the splash also travels at some speed that is much slower. By using the difference in time from the visual splash and the audio splash I can get the speed of sound. Oh, I have to use the distance too — but I just found that.

Although I’m pretty skilled with video analysis, my audio skills aren’t so great. I loaded the video in my video editor to get the time difference between the visual and audio splash.

That picture probably doesn’t really help — but I get about three frames between the two events. Since the video is recorded at 30 frames per second, these three frames would be a time interval of 0.1 seconds.

Using my height of 42.5 meters, I get a value of 421.5 m/s. That’s not too bad. The accepted value for the speed of sound at sea level and room temperature is about 343 m/s. I’m happy with this.

**Even More Questions**

Here are some other questions that I thought of during my analysis. You can answer these and the unanswered ones above as your homework assignment.

- Does it matter if he jumps UP while leaving the cliff?
- Can you get a measure of his motion by calculating his vertical position using the angular size of his body?
- What if he wants to free fall for just one more second? How much taller would the cliff need to be?