By classical consideration of motion of bodies there is no need to consider dependence of such physical quantity as body weight on change of speed, except cases of increase in body weight.
Relativistic consideration
Open the textbook in physics of the 10th class on a subject about relativistic dynamics. This section of physics describes the processes and regularities happening at motion of bodies to the speeds close to the speed of light. The fact is that when moving bodies with so high speeds some of physical quantities, considered as constants in classical physics, become dependent on speed size.
However notice that change of body weight at the movement with ultrahigh speeds is connected with great value of size of speed, but not accelerations. If you look at expression for relativistic weight, then will see that it depends on speed vector size. Acceleration of bodies in relativistic cases leads more likely to temporary shifts.
Increase in weight due to acceleration
Pay attention that at acceleration of a body in some physical cases the body weight changes. For example, it is possible to refer the movement of the person in the elevator to such cases. When the elevator begins dispersal in the direction up, the person experiences increase in body weight. In a situation when the elevator brakes, moving up, it seems to the person that his weight much less. Actually, feelings which the person in these cases has are quite valid and are easily described by classical dynamics. Draw on the sheet of paper schematically the elevator in the form of a rectangle and the person in it in the form of a point. Represent vectors of forces operating on the person at the movement in the elevator. In this case the person is affected by the gravity directed vertically down and force of reaction of a support directed up. The vector which is opposite to a vector of reaction of a support is considered body weight. The reference system connected with the elevator is not inertial therefore strength of each other is not compensated. Write down the second law of Newton, having equated the work of body weight of the person on his acceleration in the elevator to the sum of vectors of forces. From this ratio it is possible to find what force of reaction of a support is equal to. It will be equal to the work of body weight on the difference of vectors of acceleration of the elevator and acceleration of gravity. Now it is possible to pass to expression for weight, having just traded places two types of acceleration. If the elevator accelerates, moving up, then at projection of vectors of acceleration on the axis directed vertically down you receive that two types of acceleration are summarized, but are not subtracted. Thus, it turns out that at the accelerated movement of the elevator up the weight of the person increases by the size equal to the work of body weight and acceleration of the elevator. It also leads to feeling of congestion.