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 x(t) = 50.0 cm + (2.00 cm/s)t − (0.0625 cm/s2)t2. (b) At what time t is the velocity of the turtle zero?

Verified step by step guidance

1. Identify the equation for the turtle's position as a function of time, which is given by x(t) = 50.0 cm + (2.00 cm/s)t − (0.0625 cm/s^2)t^2.
2. To find when the velocity is zero, first find the velocity function by differentiating the position function x(t) with respect to time t. The velocity function v(t) is the derivative of x(t), so v(t) = dx/dt.
3. Differentiate each term of x(t) separately: The derivative of 50.0 cm is 0, the derivative of (2.00 cm/s)t is 2.00 cm/s, and the derivative of −(0.0625 cm/s^2)t^2 is −(2 * 0.0625 cm/s^2) * t.
4. Simplify the expression for v(t) to get v(t) = 2.00 cm/s - 0.125 cm/s^2 * t.
5. Set the velocity function equal to zero and solve for t: 0 = 2.00 cm/s - 0.125 cm/s^2 * t.

Key Concepts

Position Function

The position function describes the location of an object over time. In this case, the turtle's position is given by the equation x(t) = 50.0 cm + (2.00 cm/s)t − (0.0625 cm/s²)t², which includes a constant term, a linear term representing constant velocity, and a quadratic term indicating acceleration. Understanding this function is crucial for analyzing the turtle's motion.

Velocity

Velocity is the rate of change of position with respect to time and is mathematically defined as the derivative of the position function. For the turtle, the velocity can be found by differentiating the position function x(t) with respect to time t. This will yield a new function v(t) that indicates how fast and in which direction the turtle is moving at any given time.

Finding Critical Points

Critical points occur where the velocity of an object is zero, indicating a change in motion. To find when the turtle's velocity is zero, we set the velocity function v(t) equal to zero and solve for t. This process is essential for determining when the turtle stops moving forward or changes direction.