Question

A turtle crawls along a straight line, which we will call the x-axis with the positive direction to the right. The equation for the turtle’s position as a function of time is (look at the equation below).

(a) Find the turtle’s initial velocity, initial position, and initial acceleration.

(b) At what time t is the velocity of the turtle zero?

(c) How long after starting does it take the turtle to return to its starting point?

(d) At what times t is the turtle a distance of 10.0 cm from its starting point?

Answer 1

Assume we have an equation

$$x(t)=x_0+bt^2+ct.$$

(a) Initial position: $x_0$. Initial velocity: $c$. Initial acceleration: $b$.

(b) Determine velocity as the derivative of position by time:

$$v(t)=2bt+c,\\ v(t)=0=2bt+c,\\\space\\ t=\frac{-c}{2b}.$$

(c) The position is zero at:

$$bt^2+ct+x_0=0,\\ t_{1,2}=\frac{-c\pm\sqrt{c^2-4bx_0}}{2b}.$$

(d) Solve with the same equation:

$$bt^2+ct+x_0=10,\\ bt^2+ct+(x_0-10)=0,\\ t_{1,2}=\frac{-c\pm\sqrt{c^2-4b(x_0-10)}}{2b}.$$