Question

(ii)(a) An object is moving with constant acceleration in 1D (along a straight line), what is its position x(t) as a function of time, given its initial position xo, initial velocity vo and acceleration a? (b) How do you derive its velocity as a function of time from x(t)? (c) Why is the funciton of x(t) the key for predicting eclipses and hurricanes?

Answer 1

a) The acceleration can be written as rate of change of velocity

$\frac{dv}{dt}=a$

or

$dv=a\,dt$

Integrate both sides, at t = 0, v = v_o

$\int_{v_o}^vdv=a\int_0^t\,dt$

$v-v_o=at$

or

$v=v_o+at$

velocity is rate of change of position

$\frac{dx}{dt}=v=v_o+at$

or

$dx=v_o\,dt+at\,dt$

at t = 0, x = x_o

Therefore

$\int_{x_o}^xdx=v_o\int_0^t\,dt+a\int_0^tt\,dt$

$x-x_o=v_o t+\frac{1}{2}at^2$

Position as a function of time is

$x=x_o+v_o t+\frac{1}{2}at^2$

b) To derive the velocity differentiate x with respect to time

$v=\frac{dx}{dt}=v_o +\frac{2}{2}at=v_o + at$

c) If we know the acceleration of hurricanes, we can predict using the above formula beforehand where the hurricane is heading and evacuate the place to minimize the damage. Also if we know the acceleration of the moon, with respect to earth, we can predict where the eclipse can be viewed.