Final answer:
The question pertains to high school Physics, specifically projectile motion, and involves analyzing a quadratic function representing the vertical position over time. The concepts include the object's vertical motion, horizontal velocity, and the effects of gravity.
Step-by-step explanation:
The subject matter of the question revolves around the motion of a projectile, particularly analyzing its vertical position over time, which is described by the quadratic function h(t) = -2t2 + 7.86t + 5.91. This involves concepts of kinematics, a branch of physics.
The discussion of horizontal and vertical velocities, as well as the acceleration due to gravity, indicates that the student is likely dealing with a problem about projectile motion in a Physics course.
The equation h(t) = -2t2 + 7.86t + 5.91 is a quadratic equation representing the vertical motion of the projectile, which could be a ball or any other object thrown into the air. The coefficient of t2 is negative, showing that the object is being subject to gravity, which pulls it back down after it ascends.
The initial velocity can be determined from the coefficient of t, and the initial height from the constant term. Moreover, the horizontal velocity, due to the absence of any forces acting in the horizontal direction (assuming air resistance is negligible), will remain constant throughout the motion, resulting in a constant positive value on a horizontal velocity vs. time graph.
The other parts of the question that mention a quadratic equation with roots, finding a jogger's velocity using a slope, the equation of motion for a skier, and the graph of vertical velocity over time with a given slope due to gravity all pertain to calculating various aspects of motion under the influence of constant acceleration, a concept fundamental to high school Physics.