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 $x of t equals 50.0$ cm + ( $2.00$ cm/s) $t$ − ( $0.0625$ cm/s²) $t^{2}$ . Find the turtle's initial velocity, initial position, and initial acceleration.

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Identify the given position function: x(t) = 50.0 cm + (2.00 cm/s)t − (0.0625 cm/s^2)t^2. This is a quadratic equation in the form of x(t) = x_0 + v_0*t + (1/2)*a*t^2, where x_0 is the initial position, v_0 is the initial velocity, and a is the acceleration.

Determine the initial position (x_0) by evaluating the position function at t = 0. This gives x_0 = 50.0 cm.

Find the initial velocity (v_0) by identifying the coefficient of the linear term in the position function. The term (2.00 cm/s)t indicates that the initial velocity v_0 is 2.00 cm/s.

Calculate the initial acceleration (a) by identifying the coefficient of the quadratic term in the position function. The term (0.0625 cm/s^2)t^2 corresponds to (1/2)*a*t^2, so solve for a: a = 2 * 0.0625 cm/s^2.

Summarize the findings: The turtle's initial position is 50.0 cm, the initial velocity is 2.00 cm/s, and the initial acceleration is 0.125 cm/s^2.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Kinematic Equations

Kinematic equations describe the motion of objects in terms of displacement, velocity, acceleration, and time. In this problem, the position function x(t) = 50.0 cm + (2.00 cm/s)t − (0.0625 cm/s^2)t^2 is a kinematic equation that provides the turtle's position as a function of time, allowing us to extract initial conditions and understand the motion dynamics.

Initial Conditions

Initial conditions refer to the values of position, velocity, and acceleration at the start of observation (t = 0). For the turtle, the initial position is the constant term in the equation (50.0 cm), the initial velocity is the coefficient of t (2.00 cm/s), and the initial acceleration is twice the coefficient of t^2, which is -0.125 cm/s^2, derived from the equation's structure.

Differentiation in Physics

Differentiation is a mathematical process used to find rates of change, such as velocity and acceleration, from position functions. The first derivative of the position function with respect to time gives the velocity function, and the second derivative gives the acceleration. In this problem, the coefficients of t and t^2 in the position equation directly provide the initial velocity and acceleration, respectively.