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Italian physicist Galileo Galilei is usually credited with being the first to measure speed by considering the distance covered and the time it takes. Galileo defined speed as the distance covered per unit of time. [3] In equation form, that is =, where is speed, is distance, and is time. A cyclist who covers 30 metres in a time of 2 seconds ...
The general formula for the escape velocity of an object at a distance r from the center of a planet with mass M is [12] = =, where G is the gravitational constant and g is the gravitational acceleration. The escape velocity from Earth's surface is about 11 200 m/s, and is irrespective of the direction of the object.
Equation [3] involves the average velocity v + v 0 / 2 . Intuitively, the velocity increases linearly, so the average velocity multiplied by time is the distance traveled while increasing the velocity from v 0 to v, as can be illustrated graphically by plotting velocity against time as a straight line graph. Algebraically, it follows ...
For a map distance of Δx AB, the first equation above predicts a midpoint Lorentz factor (up from its unit rest value) of γ mid = 1 + α(Δx AB /2)/c 2. Hence the round-trip time on traveler clocks will be Δτ = 4(c/α)cosh −1 [γ mid], during which the time elapsed on map clocks will be Δt = 4(c/α)sinh[cosh −1 [γ mid]].
The first equation shows that, after one second, an object will have fallen a distance of 1/2 × 9.8 × 1 2 = 4.9 m. After two seconds it will have fallen 1/2 × 9.8 × 2 2 = 19.6 m; and so on. On the other hand, the penultimate equation becomes grossly inaccurate at great distances.
From the equation for uniform linear acceleration, the distance covered = + for initial speed =, constant acceleration (acceleration due to gravity without air resistance), and time elapsed , it follows that the distance is proportional to (in symbols, ), thus the distance from the starting point are consecutive squares for integer values of time elapsed.
The speed attained during free fall is proportional to the elapsed time, and the distance traveled is proportional to the square of the elapsed time. [39] Importantly, the acceleration is the same for all bodies, independently of their mass. This follows from combining Newton's second law of motion with his law of universal gravitation.
Also, the velocities in the directions perpendicular to the frame changes are affected, as shown above. This is due to time dilation, as encapsulated in the dt/dt′ transformation. The V′ y and V′ z equations were both derived by dividing the appropriate space differential (e.g. dy′ or dz′) by the time differential.