PARABOLIC FLIGHT

INTRODUCTION

Ballistic flight is flight which is only powered during the initial phase of its ascent. The major part of the trajectory is then under the influence only of gravity. This trajectory follows a parabolic path (as long as the rocket does not ascend too high).

THE TRAJECTORY

An initial rocket burn provides the initial velocity v_o at an angle θ to the horizontal . The subsequent motion is described by 3 kinematic equations (K1 to K3) with constant downward acceleration due to gravity of a = -g where g ~ 9.8 near the Earth's surface. These equations are sometimes referred to as the Galilean equations because of Galileo's interest in and experiments with this type of motion.

Applying these equations to both the x and y coordinates we have:

v_xo = v_o cos θ and v_yo = vo sin θ

and (K1) gives:

v_y = v_yo - g t

At the apogee the vertical velocity has been reduced to zero at a height of h by the vertical gravitational acceleration of -g. This allows us to determine the time to reach apogee of

t_h = v sin θ / g

And because of the symmetry of the problem, twice this time will get the missile back to the ground at a range of R

t_R = 2 v_o sin θ / g

In the x direction there is no acceleration (ignoring air resistance since the rocket travels mostly above the atmosphere) and so (K2) with a_x = 0 gives the range as:

R = v_xo t_R
= v_o cos θ t_R
= 2 v_o² sin θ cos θ / g
= v_o² sin 2 θ / g

The height is given by (K3) where v_y = 0

h = v_o² sin 2 θ / ( 2g )

To find the angle to give maximum range we compute dR/dθ and set it to 0 (for an extremum), giving cos 2 θ = 0 and θ = 45^o . Thus:

R_max = v_o² / g
h_max = R_max / 2
ToF (t_R) = √2 v_o / g (time of flight)

For true ballistic rockets only the initial velocity v_o and the initial angle θ are controllable. If a specific range is desired the angle can be varied such that:

θ = 0.5 arcsin ( g R / v_o² )

For a given range there are two solutions for θ, one where θ < 45^o and one where θ > 45^o . The lower angle has a shorter time of flight and a lower maximum height, whereas the higher angle goes higher into space and spends more time there. This is the desired solution for a sounding rocket that is sent on this suborbital trajectory to allow its payload of scientific instruments to sample the conditions along its path, both in ascent and in descent.

It should be realised that this model assumes a flat Earth, negligible air resistance over the majority of the flight, constant gravity with height and an initial impulse to produce v_o that is short in duration comparable to the time of flight. All these will alter the above results. In particular, the reduction of gravity with height will increase the maximum height, range and ToF of the missile. However, this first approximation gives us an indication of approximate rocket altitude, range and time spent in the space environment.

The table below gives range, height and time of flight for a rocket set for maximum range (θ = 45^o):

           Initial   Range    Height  ToF   Gravity
           vel km/s   (km)     (km)  (secs)  m/s/s
             1.0       102       51    144    9.5
             2.0       408      204    289    8.7
             3.0       918      459    433    7.5
             4.0      1633      816    577    6.2
             5.0      2551     1276    721    5.0

Note that even at a height of only 200 km, gravity has been reduced to about 90% of its surface value, bringing into question the original assumption that g=9.8 is constant over the altitude range. This will increase the actual height reached and the time of flight.

Australian Space Academy