Question #10a38
1 Answer
(see explanation, specifically the bottom part for a brief overview)
Explanation:
At time
The velocity as a function of time is represented by
and the acceleration is
(the acceleration is not constant, but its rate of change with time is, since its equation is that of a line)
A graph of position over time:
graph{2x^3 - 21x^2 + 60x + 3 [-10, 20, -10, 100]}
The particle will change direction when its instantaneous velocity is equal to
which agrees with the graph.
Here's a graph of the parabolic velocity equation to prove this:
graph{6x^2 - 42x + 60 [-5, 7.5, -20, 40]}
The positions at which the velocity is
and
and its initial velocity is
Let's now look at how the velocity changes over time, by examining its acceleration:
In the velocity-time graph, if the slope of the tangent line of the parabola at a given time is negative, the acceleration is negative, and if the slope of the tangent line is positive, it's acceleration is positive. Thus, the particle's acceleration is negative (it's slowing down) from time
Which is what the graph shows.
In short, from time