When gravity is constant and upward is taken as positive, the vertical acceleration is downward and equal to -g. With constant acceleration, velocity changes linearly in time: v_y(t) = v_y0 + a_y t. Substituting a_y = -g gives v_y(t) = V0y - g t. This form matches the initial condition that at time zero the vertical velocity is V0y, and it shows that the upward velocity decreases by g each second due to gravity, eventually reaching zero at the top of the trajectory and becoming negative as the object falls. The other expressions don’t fit the situation: one would imply gravity is acting upward, which is incorrect; another would flip the initial sign of the velocity; and the last uses t^2, which is appropriate for displacement, not velocity (the displacement is y = y0 + V0y t - (1/2) g t^2).

Sound intensity is the power per unit area—the rate at which energy passes through a given area. If a source emits power P and that power spreads over an area A, the intensity is I = P/A. This makes intuitive sense: more area means the same amount of power is spread thinner, so the intensity goes down in proportion to 1/A. In free space, a point source radiates over a spherical area A = 4πr^2, giving I = P/(4πr^2). The units are watts per square meter. Expressions that multiply by area or square the power don’t match the idea of energy crossing area, which is why they aren’t correct.

Speed is a scalar quantity that measures how fast something is moving, giving only the magnitude of motion. Because it describes how much motion, not where it’s headed, speed has no direction. That’s why the blank is filled with “direction.” And since speed is the magnitude of motion, it’s taken as a positive value (in practice nonnegative; you can have zero when at rest, but we usually express speed as a positive quantity when motion is occurring). In contrast, velocity includes direction and can be positive or negative depending on the chosen frame or axis, and the other options mix in properties that speed doesn’t possess (like a definite magnitude being negative, or equating speed with velocity, or tying it to a reference frame).

Wavelength is the spatial period of a wave—the distance over which the wave’s pattern repeats. The distance between successive crests (or between successive troughs) is the wavelength because after you move that amount along the direction of travel, the wave looks the same again. Crest is the maximum displacement, which is a height measure, not a spacing; that height is called the amplitude when measured from the middle of the wave. Trough is the minimum displacement, also a height measure, not spacing. So the first statement correctly defines what wavelength is.