We present the results of numerical calculations of the influence of air drag force, quadratic in speed, on the values of such characteristics of projectile motion as maximum height, maximum range of projectile, total flight time, final speed and two parameters which characterize the asymmetry of flight trajectory. It is found that deviations from the idealized case of free projectile motion for one part of these characteristics increase with increasing of the initial angle of throwing, whereas for the other they decrease. The greatest deviations are observed for maximum range. We also consider the properties of optimal launch angle and low- and high-angle firing. We conclude that such predictions as the asymmetry of the projectile trajectory, the decrease in the total flight time and the optimal launch angle in comparison with the values of these quantities for the case of free projectile motion, as well as the peculiarities of low- and high-angle firing, can only be made on the basis of numerical analysis.

In this paper we search the shape of an aspherical body and the direction in space, for which the greatest deviations from the point mass field (the difference from the inverse-square law) take place for large distances from the field source. It turns out to be a system of two equal point-like masses at the poles of a fixed sphere (giving the greatest positive deviations from the point mass field) and uniform distribution of point-like masses (discrete or continuous) around the sphere equator (giving the greatest negative deviations from the point mass field). In these cases the extremal direction of the field measurement respectively passes through point-like particles and coincides with the axis of symmetry of a ring, which is perpendicular to its plane. Our numerical estimations show that any body can be considered with reasonable accuracy (the relative error in the determination of the field strength is less than $5 \%$) as point-like mass if the distance to the observation point is more than an order of magnitude larger than its characteristic sizes. The problem considered in this paper can help readers to probe the limits of applicability of the field point source model.

In this paper, we analyse the process of a particle sliding down an arbitrary curved hill in the presence of friction. We show that in this case, the kinetic friction force turns out to be effectively dependent on the speed. We find that the final speed for each concave (convex) profile should be less (greater) than that for the corresponding inclined plane. In the limit of extremely small final speed or profiles of the hill close to linear, the final speed does not depend on the shape of the trajectory even in the presence of kinetic friction. As an illustrative example, we perform the calculation of the final speed for a concave hill in the form of a quarter circle. We also derive the shape of the curved hill profile for which a body can slide down at a constant speed.

We discuss the influence of two factors on the deviations from the model of the magnetic field of a long straight wire. Firstly, we derive the magnetostatic analogue of McCullagh’s formula for the case of the infinitely long straight conductor with an arbitrary cross-section. Using this equation on the basis of its practical significance, we find that the greatest deviations from the long straight wire model should be observed for a system of two parallel and infinitely long straight wires, equal in absolute value and direction currents. In this case the extremal direction of the magnetic field deviations respectively passes through these wires or is perpendicular to the plane passing through them. Our numerical estimations show that any infinitely long straight conductor with a cross-section having dihedral group symmetry can be considered with reasonable accuracy (the relative error in the determination of the magnetic flux density is less than 5%) as the infinite straight wire, if the distance to the observation point is approximately more than half order of magnitude larger than its maximum transverse size. We also analyse the effect of finite length of the carrying wire on the magnitudes of the magnetic field. We show that in this case, the smallest deviations from the infinite straight wire model should be observed for points belonging to the perpendicular plane passing through the midpoint of a wire.

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We consider the cases of a beam hanging from four and three strings. In addition to the static equilibrium equations, these statically indeterminate structures can be easily resolved for the unknown tension forces with only using Hooke’s law.

The point particle is an idealized object where rotational and vibrational motion is ignored. Nevertheless, in many cases such degrees of freedom play a significant role. For example, the rotation and vibration of a molecule is an important “reservoir” of its internal energy. The excitation of these types of motion can occur during the collision process. Taking into account this effect, the collisions can no longer be considered as perfectly elastic but actually are inelastic. Some models for inelastic collisions have already been discussed and compared to experimental results. Here we present a mechanical system wherein we describe the vibrations upon impact in perhaps the simplest way possible; also we provide a qualitative explanation of the collision of a mono-atomic molecule with a nonrigid diatomic molecule.

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