Explain the mathematical and graphical relationship between velocity and acceleration using your own words. 0 Word(s) 2. Explain the mathematical and graphical relationship between velocity and position using your own words. 0 Word(s) 3. How does the angle of inclination affect the velocity of the marble

1) a = dv / dt , v = v₀ + a t

2) v = dx / dt,   x = v₀ t + ½ a t²

Explanation:

This is an exercise in kinematics

1) they ask us for the relationship between speed and acceleration

 general case

                     a = dv / dt

constant acceleration case

                     v = v₀ + a t

these are the mathematical relationships between acceleration and speed, we see that for cases where acceleration is constant the relationship is a straight line, so it is very easy to work, when we have several accelerations in a problem, we define a different range for each and we work as if they were separate problems.

A graphical relationship is the line where the slope is the value of the acceleration.

2) The mathematical relationship between speed and position is

non-constant acceleration case

              v = dx / dt

case with zero acceleration

          v = x / t

case with nonzero acceleration

          x = v₀ t + ½ a t²

we see that for this part we have more possibilities when there is no acceleration the relationship is a straight line and when there is acceleration it is a parabola. Again each interval to work is determined by the value of the acceleration

In a graphical representation we have two possibilities, a line where the slope of the initial velocity and a parabola

3) The angle of inclination affects speed because it has two parts, one on the x-axis and one on the y-axis, each one less than the initial speed. These speeds are given by

         vₓ = v₀ cos θ

         v_y = v₀ sin θ

and the initial velocity is

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