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/s^2)t^2. (a) Find the turtle's initial velocity, initial position, and initial acceleration.

Verified step by step guidance

1. Identify the coefficients of the position equation x(t) = 50.0 cm + (2.00 cm/s)t − (0.0625 cm/s^2)t^2. The equation is in the form of a quadratic equation x(t) = x_0 + v_0t + (1/2)at^2.
2. Recognize that the constant term in the equation, 50.0 cm, represents the initial position x_0 of the turtle.
3. Identify the coefficient of the t term, 2.00 cm/s, which represents the initial velocity v_0 of the turtle.
4. Note that the coefficient of the t^2 term, -0.0625 cm/s^2, is half of the initial acceleration. Therefore, to find the initial acceleration a, multiply this coefficient by 2.
5. Summarize the initial conditions: initial position x_0 is the constant term, initial velocity v_0 is the coefficient of t, and initial acceleration a is twice the coefficient of t^2.

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/s^2)t^2, where the first term represents the initial position, the second term indicates the linear change in position due to velocity, and the third term accounts for the quadratic change due to acceleration.

Initial Velocity

Initial velocity refers to the speed and direction of an object at the start of observation, typically at time t=0. In the turtle's position function, the coefficient of the linear term (2.00 cm/s) represents the initial velocity, indicating how fast the turtle is moving along the x-axis at the beginning of its motion.

Initial Acceleration

Initial acceleration is the rate of change of velocity at the start of the motion. In the given position function, the coefficient of the quadratic term (-0.0625 cm/s^2) represents the acceleration of the turtle. A negative value indicates that the turtle is decelerating as it moves along the x-axis.