## 1 Introduction  
Improvised explosive devices (IEDs) have caused harm to hundreds of thousands of people, with over 170,000 people affected in the last decade (2010-2020) [1]. They are especially damaging and lethal because they are often embedded with solid objects, these being normally readily available items such as ball bearings or nails. These embedded objects become projectiles, causing greater fatality rates, traumatic amputations, and soft tissue injuries from IEDs [2, 3]. Much research is being conducted to detect IEDs [4, 5] and prevent their detonation [6, 7]. Other approaches include improved armour for military targets [8], hardening of large-scale critical civilian infrastructure against large IED attacks [9], and research into the destruction of IEDs in the battlefield arena [7]. [PERSON] et al. [10] report on the extraction of touch DNA and fingerprints from detonated IEDs to facilitate the tracking and prosecution of terrorists. These advances are encouraging, but it is unlikely that they will entirely prevent future detonations of person-borne IEDs used in terror attacks. Understanding the behaviour of blast-driven ball bearings embedded in explosive charges will aid in the development of blast protection from IEDs. One major challenge in IED research is the huge range of potential IED geometries, sizes, explosion types, and embedded object configurations. Hence, a fundamental understanding of the momentum transfer mechanisms of embedded solid particles from the explosive detonation is needed. While there is a large body of literature on cratering damage due to impact, research into blast-induced projectile impacts is focused on fragmentation [11; 12; 13; 14], such as that in warhead or blast-driven metal [15]. A limited experimental study on the damage caused by incorporating a foreign object in an explosive charge is available in reference [16]. [PERSON] and [PERSON] [16] did not measure the velocity of the explosively driven projectiles but analysed the damage distribution in the walls of a surrounding steel cylinder. They used the penetration depth to infer information about the effect of packing pattern. In recent work by [PERSON] et al. [17], a novel experimental arrangement for gaining fundamental insights into the momentum transfer and flight of a solid insert from a simplified IED was proposed and developed. The experiments and subsequent numerical work focussed on the flight characteristics of a single ball bearing driven by the detonation of a plastic explosive. Cylindrical charges were studied as these were deemed more practical and realistic geometries for IEDs. The effect of charge shape has been shown to be highly influential in the near field, while its influence on peak pressure was almost insignificant in the far field when the Hopkinson-Cranz scaled distance exceeded 5 m/kg\({}^{1/3}\)[18]. For IEDs, the charge shape is a critical factor. Rear detonated cylindrical charges have been shown to generate the highest axial impulses and pressures [19], meaning that they would be expected to produce the highest ball bearing velocities (therefore a worst-case scenario for an IED threat). The influence of the charge aspect ratio on flight characteristics was investigated by [PERSON] et al. [17]. The charge mass was varied but the ball bearing size and mass was kept constant. Aspect ratios below a critical value were more efficient at transferring momentum to the ball bearings. Some preliminary remarks regarding the damage to some of the ball bearings were reported by [PERSON] et al. [20]. This paper reports the results of experimentally informed computational simulations that examine the influence of ball bearing size on the velocity and damage characteristics of an explosively driven single ball bearing. The paper provides detailed insights into the flight characteristics of a ball bearing embedded in a rear detonated cylindrical charge and how its velocity varies with ball bearing mass and diameter, which is of practical relevance to IED research.