Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 19.6 m/s from a height of 24.5 m above the ground. The height (in meters) of the stone above the ground t seconds after it is thrown is s(t) = -4.9t² + 19.6t + 24.5.
On what intervals is the speed increasing?

검증된 단계별 안내

Step 1: Understand that the speed of the stone is increasing when the magnitude of its velocity is increasing. The velocity function v(t) is the derivative of the height function s(t).

Step 2: Differentiate the height function s(t) = -4.9t^2 + 19.6t + 24.5 to find the velocity function v(t). This gives v(t) = s'(t) = -9.8t + 19.6.

Step 3: The speed is increasing when the derivative of the velocity, which is the acceleration, has the same sign as the velocity. Calculate the acceleration by differentiating the velocity function: a(t) = v'(t) = -9.8.

Step 4: Since the acceleration a(t) = -9.8 is constant and negative, the velocity is decreasing when it is positive and increasing when it is negative. Find when v(t) = 0 to determine when the velocity changes sign.

Step 5: Solve the equation v(t) = -9.8t + 19.6 = 0 to find the critical point where the velocity changes sign. This will help determine the intervals where the speed is increasing.

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주요 개념

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

Velocity and Speed

Velocity is a vector quantity that refers to the rate of change of position with respect to time, including direction. Speed, on the other hand, is the magnitude of velocity and does not consider direction. In this context, understanding how velocity changes over time is crucial for determining when the speed of the stone is increasing.

Acceleration

Acceleration is the rate of change of velocity with respect to time. In the case of the stone thrown upward, the acceleration due to gravity is acting downward at approximately -9.8 m/s². Analyzing the acceleration helps determine whether the speed of the stone is increasing or decreasing, particularly when the velocity changes sign.

Critical Points and Intervals

Critical points occur where the derivative of a function is zero or undefined, indicating potential maxima, minima, or points of inflection. To find intervals where speed is increasing, one must analyze the derivative of the velocity function, identifying where it is positive. This involves determining the intervals on the time axis where the speed of the stone is increasing based on the sign of the derivative.