Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

문제 4.9.109c

4장, 문제 4.9.109c

107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .
c. Find the time when the object reaches its highest point. What is the height?
A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.

검증된 단계별 안내

Step 1: Start with the given acceleration equation:

a (t) = - g

, where

g

is the acceleration due to gravity (9.8 m/s²). Integrate this equation to find the velocity function

v (t)

. The integration gives

v (t) = - g t + v

₀, where

v

₀ is the initial velocity.

Step 2: Substitute the initial velocity of the payload, which is given as 10 m/s (the rate at which the hot-air balloon is rising), into the velocity equation. This gives

v (t) = - 9.8 t + 10

Step 3: To find the time when the object reaches its highest point, set the velocity

v (t)

equal to zero (since the object momentarily stops moving upward at its highest point). Solve the equation

- 9.8 t + 10 = 0

for

t

Step 4: Once the time

t

is found, use the position equation to find the height at this time. The position equation is obtained by integrating the velocity equation:

s (t) = - \frac{1}{2} g t^{2} + v

₀t+s₀, where

s

₀ is the initial elevation (400 m).

Step 5: Substitute the values for

g

v

₀,

s

₀, and the time

t

(found in Step 3) into the position equation to calculate the height of the object at its highest point.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

Acceleration due to Gravity

Acceleration due to gravity, denoted as 'g', is a constant that represents the rate at which an object accelerates towards the Earth when in free fall. On Earth, this value is approximately 9.8 m/s². In the context of vertical motion, it is crucial for determining how quickly an object's velocity changes as it moves upward or downward.

Velocity and its Relationship to Acceleration

Velocity is the rate of change of an object's position with respect to time, and it can be affected by acceleration. In this scenario, the object's initial velocity is given as 10 m/s (upward). Understanding how to apply the acceleration due to gravity to this initial velocity is essential for calculating the object's motion and determining when it reaches its highest point.

Maximum Height in Projectile Motion

The maximum height of an object in projectile motion occurs when its velocity becomes zero before it starts descending. To find this height, one can use kinematic equations that relate initial velocity, acceleration, and displacement. In this case, the object released from the balloon will rise until the upward velocity is countered by the downward acceleration due to gravity.