When a projectile of mass m is launched at an angle θ (wrt to horizontal ) and an initial speed v_o, it is subjected to gravitational force in the vertical (towards the ground) direction and zero force in the horizontal direction. The distance travelled is given by –

$$ x = v_o cos\theta *t \\ y = \frac{1}{2}gt^2 + v_o sin\theta * t $$

In the experiment, θ and x are easily measured. It is interesting to compare the experimentally measured range with the one calculated with theory. The values of v_o and t, which are cannot be measured easily, are needed to calculate the range. Time of flight is very small and comparable to the human error for the measurement of time. Also it is not possible to measure v_o directly. However, by launching the projectile at θ=0° , and using the above equations, with H as height of the launch and R° the range -

$$ H = \frac{1}{2}gt^2 \rightarrow t = \sqrt{\frac{2H}{g}} \\ R_o = v_o \sqrt{\frac{2H}{g}} \rightarrow v_o = R_o \sqrt{\frac{g}{2H}} $$

This value of v_o remains the same for all the angles for a particular position of the spring. For θ>0, solve the quadratic equation for vertical motion to obtain the value of t. Use this value of t to calculate the range.