Cambridge Engineering Interview: Horizons and Escape Velocity

Engineering

Madeleine S. · Engineering (Jesus College, Cambridge)

Hi, I'm Madeleine, and I've just finished my second year of Engineering at Jesus College, Cambridge. I applied in 2012 and started in 2013. I want to walk through two problems from I Want to Study Engineering (i-want-to-study-engineering.org), a free bank of interview-style questions run by Cambridge's Department of Engineering. Both are close to what I was asked in my own interviews, and both reward the same habit: draw the picture, find the one principle that matters, and only then do the algebra.

The horizon question

The first question goes like this. On a clear day, you're on a plane 38,000 feet above the middle of the Pacific Ocean. Taking the radius of the Earth as 6,400 km, roughly how far away is the horizon? You're told that 1 foot is 0.3048 metres.

Start with the units: 38,000 times 0.3048 is about 11,600 metres, so the plane is roughly 11.6 km up. Then draw the Earth as a circle, mark its centre, and put the plane just above the surface. The wording helps too: "on a clear day" and "the middle of the Pacific" mean no haze and no mountains, so you can treat the Earth as a smooth sphere.

Why the right angle matters

The horizon is the furthest point on the sea you can see, so your line of sight just grazes the surface. That makes it a tangent to the circle, and a tangent always meets the radius at a right angle. It's easy to call the right angle obvious, but giving the reason is what turns a guess into an argument, and interviewers notice the difference.

Now you have a right-angled triangle. One side is the radius, 6,400 km. The hypotenuse runs from the centre of the Earth to the plane, 6,411.6 km. The third side is the distance to the horizon, d. By Pythagoras, d squared equals 6,411.6 squared minus 6,400 squared, so d is roughly 386 km.

A shortcut worth knowing

Before reaching for the calculator, expand the bracket. If R is the radius and h the height of the plane, d squared equals 2Rh + h squared. Because the plane is so close to the surface, h squared is tiny next to 2Rh, about 135 against nearly 150,000. Drop it, and d is roughly the square root of 2Rh, which gives 385 km, within half a kilometre of the exact answer.

That formula tells you more than the number does. The distance to the horizon grows with the square root of your height, so going four times higher only lets you see twice as far. Standing on a beach with your eyes 1.7 metres above the sea, the horizon is only about 4.7 km away.

The rogue planet question

The second question: a rocket of mass m is to be launched from the surface of a rogue planet with mass M, radius R and no atmosphere. By making reasonable assumptions about the distance between the planet and any nearby galaxies, find the escape velocity the rocket needs to overcome the planet's gravitational field.

"Reasonable assumptions" is an invitation to decide what to ignore. A rogue planet doesn't orbit any star, and the mention of galaxies points to the key assumption: if other planets or stars were nearby, their pull would affect the rocket too, so you assume everything else is so far away that only this planet's gravity matters. You can add a few more. No atmosphere means no drag, a spherically symmetric planet pulls as though all its mass were at its centre, and escape velocity assumes the rocket gets all its speed at launch, like a cannonball.

Using energy instead of forces

Draw the planet, mark the rocket at a distance r from the centre, and write down Newton's law of gravitation: the force on the rocket is GMm divided by r squared. If you can't remember that formula, just say so; the interviewers will usually give it to you, because they don't want you stuck.

You might reach for F = ma next, but the force weakens as the rocket climbs, so the deceleration isn't constant and the suvat equations don't apply. When a force changes with distance, think about energy instead.

Work done is force times distance, but only for a constant force, so split the journey into tiny steps of length delta r, over which the force barely changes. Adding up the work for every step from the surface out to infinity gives the integral of GMm divided by r squared, from R to infinity. Writing one over r squared as r to the minus 2, this integrates to minus GMm divided by r, which is zero at infinity, so the total work done is GMm divided by R. If this step doesn't come to you in the interview, don't panic; it's exactly where interviewers tend to give you a nudge.

The result and how to check it

The work done has to come from the rocket's kinetic energy at launch, so one half m v squared equals GMm divided by R. Rearranging gives the escape velocity: v equals the square root of (2GM divided by R).

Three quick checks make the answer convincing. The rocket's mass cancels, so a pebble and a battleship need the same launch speed; escape velocity depends only on the planet. The units come out as metres per second. And since g equals GM divided by R squared, the answer can also be written as the square root of 2gR. For Earth, with g = 9.81 m/s squared and the same 6,400 km radius as the first question, that gives about 11.2 km/s, the familiar value.

There's a deeper point too. Gravity never switches off, yet the work needed to escape is finite, because a force that falls off as one over r squared fades quickly enough for the integral to settle on a finite value. If gravity fell off as one over r, nothing could ever escape.

What the interviewers are really testing

Neither question needs more than school maths and physics, but both ask you to use them in ways you probably haven't been shown. From my own interview, and from talking to my interviewers who are now my supervisors, the aim is to see how quickly you pick up new ideas, at least at my college. It feels a lot like a Cambridge supervision. They're looking for:

• turning a worded problem into a clear diagram before calculating anything
• giving a reason for each step, like why the line of sight meets the radius at a right angle
• stating sensible assumptions and explaining why they're reasonable
• choosing the right principle, such as energy conservation, when a force changes
• picking up a hint and running with it

Common mistakes to avoid

Watch out for reaching for the calculator before drawing a diagram, asserting the right angle without saying why, and using suvat equations when the acceleration isn't constant. The biggest mistake is going silent when you're stuck: if you forget an equation, misremember one, or can't see where a question is going, say so.

Final thoughts

Try not to panic. The interview is stressful, but the interviewers are there to help, and they won't leave you stuck on a question. Don't be afraid to admit when you don't know something; I misquoted F = ma in my interview and still got in. Draw the picture first, explain your reasoning out loud, and check your answer against something you already know. There are plenty more problems like these on the I Want to Study Engineering website. Try to enjoy the interview, too: you'll be talking to some of the brightest academics in engineering, and they're not the enemy. If you'd like help preparing for your Cambridge Engineering interview, Oxbridge Solution can support you.