allow this, and Robins' mathematical approach was slightly more complex. He used the period of oscillation and mass of the pendulum (both measured with the bullet included) to calculate the rotational inertia of the pendulum, which was then used in the calculations. Robins also used a length of ribbon, loosely gripped in a clamp, to measure the travel of the pendulum. The pendulum would draw out a length of ribbon equal to the chord of pendulum's travel. The first system to supplant ballistic pendulums with direct measures of projectile speed was invented in 1808, during the Napoleonic Wars and used a rapidly rotating shaft of known speed with two paper disks on it; the bullet was fired through the disks, parallel to the shaft, and the angular difference in the points of impact provided an elapsed time over the distance between the disks. A direct electromechanical clockwork measure appeared in 1848, with a spring-driven clock started and stopped by electromagnets, whose current was interrupted by the bullet passing through two meshes of fine wires, again providing the time to traverse the given distance. Mathematical derivations Most physics textbooks provide a simplified method of calculation of the bullet's velocity that uses the mass of the bullet and pendulum and the height of the pendulum's travel to calculate the amount of energy and momentum in the pendulum and bullet system. Robins' calculations were significantly more involved, and used a measure of the period of oscillation to determine the rotational inertia of the system. Simple derivation We begin with the motion of the bullet-pendulum system from the instant the pendulum is struck by the bullet. Given , the acceleration due to gravity, and , the final height of the pendulum, it is possible to calculate the initial velocity of the bullet-pendulum system using conservation of mechanical energy (kinetic energy + potential energy). Let this initial velocity be denoted by . Suppose the masses of the bullet and pendulum are and respectively. The initial kinetic energy of the system Taking the initial height of the pendulum as the potential energy reference , the final potential energy when the bullet-pendulum system comes to a stop is given by So, by the conservation of mechanical energy, we have: Solve for velocity to obtain: We can now use momentum conservation for the bullet-pendulum system to get the speed of the bullet, , before it struck the pendulum.