Suppose your car is travelling at 72 km/h, which is 20 metres per second. We can also write this as 20 m/s (or if you know enough mathematics, 20 ms-1). If you reach a motorway and accelerate in 10 seconds to 108 km/h, or 30 m/s, then in each of those 10 seconds, you increase your speed by one metre per second, every second. Physicists say your acceleration is one metre per second each second, which means one metre per second per second, or 1 ms-2.
With a value of g (the acceleration due to gravity) of 9.8 ms-2, a falling object increases its velocity by 9.8 ms-1, each second. Got it? If not, don’t worry. Still, one of the problems I played with as a schoolboy was, “if you jumped from the Empire State Building, would you penetrate the roadway or splatter?”
A table of theoretical speeds, generated in a spreadsheet.
Lacking sufficient data, my friends and I agreed on the
compromise that you would splatter as you went through (well, we were schoolboys and physics students). The velocity v is given from v2 = 2as, where a is the
acceleration due to gravity and s is the distance in metres. We should use v2 = u2 + 2as, but we assume that u, the
initial velocity, is zero, so the simpler formula works.
The elapsed time
is calculated from v = u + at, but
once again, u is zero, so we fiddle v =
at to t = v/a, and use this to
get the time.
Now look up terminal velocity to work out why this table is wrong, or at the very least, misleading.
Now it gets hairy: those under about 16, STOP HERE!
Right, now it's just us oldsters and all the determined youngsters who ignore warnings (I was always one of those), I want to talk about escape velocity and black holes.
I will start with escape velocity. The
value of this for a projectile leaving a planet’s surface is equal to the
square root of 2GM/R, where G is the universal gravitational
constant, M is the mass of the planet or other body, and R is the distance from
the surface to the centre of gravity.
Some typical escape velocities.
If you want to fire a gun into space from sea level, and
send a bullet so far away that it would never fall back again, the bullet has
to leave the muzzle at about 11.2 kilometres a second. The same gun would only
have to send the bullet off at 2.4 kms-1 from the Moon or 5.0 kms-1
from Mars, but Jupiter’s escape velocity is 59.5 kms-1. The heavier
a body, the higher the escape velocity.
Of course, if the Sun turned into a neutron star and
collapsed down to almost nothing, R would be much less, and the escape velocity
from the surface would be much higher. A mountain with a mass of 1011
kg would also be a black hole, if it could be compressed down to a radius of 10–13
cm, about the size of a single proton. This might be a challenge, but the Sun,
compressed sufficiently, would have an escape velocity of c, the speed of light. (Remember c, because we will meet it again, and again and again…)
If the Earth could be compressed into a sphere just 1 cm
across, while still retaining its mass of ~1024 kg, the escape
velocity would now be equal to the speed of light, and the Earth would become a
black hole. Now about those “light particles”: in the late 1700s, everybody
thought light was corpuscular, meaning it came in the form of particles that
were subject to gravitational pull. The argument about waves and particles
would, in the end, give us quantum physics, but we aren’t there, yet.
Throw in a bit of simple Newtonian theory, and you might
have a really large, heavy star, where the escape velocity was more than the
speed of light. Even if the star glowed inside, Michell and Laplace said, no
light could ever escape to be seen.
That's enough for one day.
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