Thinking Better by Marcus du Sautoy

Author:Marcus du Sautoy [Sautoy, Marcus du]
Language: eng
Format: epub
Publisher: HarperCollinsPublishers
Published: 2021-07-05T17:00:00+00:00

A universe in flux

Even before John Glenn had completed his orbits of the Earth, calculus helped get him up there. As he sat on the launch pad he knew the spaceship would need to achieve a particular speed to be able to clear the gravitational pull of the Earth, called the escape velocity. But knowing what the speed of a spaceship is at any point as it is propelled into space is not an easy task. Things are constantly changing: the mass of the craft is shrinking as it burns off fuel, the pull of gravity is decreasing as it travels further and further away from Earth. As the push of the jets and the pull of gravity compete, it seems an impossible puzzle to unpick. But the real strength of calculus is that it can accommodate a hugely complex range of changing variables to give a snapshot of what is happening at any particular moment in time.

And it all began with that apple falling from the tree in Newton’s garden in Woolsthorpe Manor in Lincolnshire. He had retreated back to his family home from his college in Cambridge after plague had hit. Lockdown during a pandemic has certainly been a productive time for some. Shakespeare is said to have completed King Lear while the Globe was closed during a lockdown. As Newton sat in his garden, he wanted to make sense of the challenge of calculating the speed of his apple at any point on its journey from the tree to the ground. Speed is distance travelled divided by the time it takes to travel that distance. That’s fine if the speed is constant. But the trouble was that because of the pull of gravity the speed was constantly changing. Any measurements that Newton did would only give him the average speed over the period of time he was measuring.

To get a better calculation of the speed he could take smaller and smaller time intervals. But to get the exact speed at any moment really means taking an infinitely small time interval. Ultimately you want to divide distance by zero time. But how do you divide by 0? Newton’s calculus made sense of this.

Galileo had already discovered the formula for finding out how far the apple had fallen after any length of time. After t seconds the apple falls a distance of 5t2 metres. The 5 here is a measure of the particular pull of gravity on Earth. An apple tree on the Moon will have a smaller number in the equation because its gravity is smaller and the apple falls more slowly. Glenn’s spacecraft would have to keep track of this number changing as he got further from the Earth.

Let’s take the apple and throw it directly up in the air. I’m going to launch it from my hand at a speed of 25 metres per second. Baseball pitchers can hit speeds of over 40 metres per second so this isn’t unreasonable. The formula for how high the ball is from my hand after launch becomes 25t − 5t2.

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