Assume the acceleration of the object is a(t) - -9.8 meters per second. (Neglect air resistance.) With what initial velocity must an object be thrown upward (from a height of 3 meters) to reach a maximum height of 360 meters? Step 1 We want to find the initial velocity, v(0) of an object thrown upward such that the maximum of the height function (t) is 360. We also assume the acceleration due to gravity is a(t) = -9.8 meters per second. First, we must find an equation for the velocity and height functions. Recall that the derivative of the velocity function is acceleration, or v(t) - a(t). This also means that v(t) is the antiderivative of the constant function a(t). Find v(t). v(t) = ) dt -9.8 dt -9.8 t +C The initial velocity is the velocity at time t0. Find the initial velocity by substituting 0 for t in the equation above. v(0) --9.8 Step 2 We have found that the velocity of the object thrown upward has the general solution v(t) - -9.8t + C, and the initial velocity is v(0) - C. Substituting v(0) for the constant C to find the particular solution for the velocity gives us the following. v(t) - -9.8t + v(0) Now recall that the derivative of the height function is the velocity, or s(t) = v(t). This also means that s(t) is the antiderivative of v(t). Find s(t). s(t) - vit) at -(-9.8t + v(0)) at e?+ v(0)e + C,
Assume the acceleration of the object is a(t) - -9.8 meters per second. (Neglect air resistance.) With what initial velocity must an object be thrown upward (from a height of 3 meters) to reach a maximum height of 360 meters? Step 1 We want to find the initial velocity, v(0) of an object thrown upward such that the maximum of the height function (t) is 360. We also assume the acceleration due to gravity is a(t) = -9.8 meters per second. First, we must find an equation for the velocity and height functions. Recall that the derivative of the velocity function is acceleration, or v(t) - a(t). This also means that v(t) is the antiderivative of the constant function a(t). Find v(t). v(t) = ) dt -9.8 dt -9.8 t +C The initial velocity is the velocity at time t0. Find the initial velocity by substituting 0 for t in the equation above. v(0) --9.8 Step 2 We have found that the velocity of the object thrown upward has the general solution v(t) - -9.8t + C, and the initial velocity is v(0) - C. Substituting v(0) for the constant C to find the particular solution for the velocity gives us the following. v(t) - -9.8t + v(0) Now recall that the derivative of the height function is the velocity, or s(t) = v(t). This also means that s(t) is the antiderivative of v(t). Find s(t). s(t) - vit) at -(-9.8t + v(0)) at e?+ v(0)e + C,
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 94E
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Transcribed Image Text:Assume the acceleration of the object is a(t) = -9.8 meters per second. (Neglect air resistance.)
With what initial velocity must an object be thrown upward (from a height of 3 meters) to reach a maximum height of 360 meters?
Step 1
We want to find the initial velocity, v(0) of an object thrown upward such that the maximum of the height function (t) is 360. We also assume the acceleration due to gravity is a(t) = -9.8 meters per second.
First, we must find an equation for the velocity and height functions. Recall that the derivative of the velocity function is acceleration, or v(t) = a(t). This also means that v(t) is the antiderivative of the constant function a(t).
Find v(t).
v(t) =
-9.8 dt
- (-9.8
-9.8 t + C
The initial velocity is the velocity at time t- 0. Find the initial velocity by substituting 0 for t in the equation above.
v(0) =-9.8
Step 2
We have found that the velocity of the object thrown upward has the general solution v(t) = -9.8t + C, and the initial velocity is v(0) - C.
Substituting v(0) for the constant C to find the particular solution for the velocity gives us the following.
v(t) - -9.8t + v(0)
Now recall that the derivative of the height function is the velocity, or s'(t) - v(t). This also means that s(t) is the antiderivative of v(t).
Find s(t).
s(t) =
vtt) at
-9.8t + v(0)) dt
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