Answer:
a. The kinetic energy of the 2-kg steel ball as a function of velocity forms a parabolic curve. The standard equation for a parabola in vertex form is y = a(x-h)^2 + k, where (h,k) is the vertex. In this case, the vertex represents the point where the ball has the maximum kinetic energy. To find the vertex, we need to identify the value of velocity that gives the maximum kinetic energy. From the graph, we can see that the maximum kinetic energy is 104 J and it occurs at a velocity of 4 m/s. Therefore, the vertex is (4, 104).
To find the focus and directrix of the parabola, we need to determine the value of the parameter 'p', which is the distance between the vertex and the focus or directrix. The focus is a point on the axis of symmetry that is equidistant from the vertex and the directrix. The directrix is a line that is perpendicular to the axis of symmetry and is located on the opposite side of the vertex.
The value of 'p' can be found using the equation p = 1/(4a), where 'a' is the coefficient of the squared term in the vertex form equation. In this case, 'a' is negative since the parabola opens downwards. From the graph, we can estimate that the value of 'a' is approximately -2/25. Therefore, p = 1/(4*(-2/25)) = -25/8.
Since the parabola opens downwards, the focus is located below the vertex at a distance of 'p' along the axis of symmetry. Therefore, the focus is at (4, 104 - 25/8) = (4, 99.125). The directrix is a horizontal line that is located 'p' units above the vertex. Therefore, the directrix is at y = 104 + 25/8 = 109.125.
b. We are given that the car bumper has to withstand an impact of 25 J from the 2-kg steel ball without any damage. We can use the kinetic energy equation KE = 1/2mv^2 to solve for the velocity required to produce this amount of energy. Rearranging the equation, we get v = sqrt(2KE/m). Substituting the values of KE = 25 J and m = 2 kg, we get v = sqrt(2*25/2) = 5 m/s.
Therefore, the 2-kg steel ball needs to be moving at a velocity of 5 m/s when it strikes the bumper in order to produce an impact of 25 J without any damage.