Questions: The position of a hummingbird flying along a straight line is given by the functions s(t)=2t^4-2t^3-2t^2-8 where t is measured in seconds and s is measured in meters. Find the acceleration function a(t).
Provide your answer below:
a(t)=
Transcript text: Question
The position of a hummingbird flying along a straight line is given by the functions $(t)=2 t^{4}-2 t^{3}-2 t^{2}-8$ where $t$ is measured in seconds and $s$ is measured in meters. Find the acceleration function $a(t)$.
Provide your answer below:
\[
a(t)=
\]
$\square$
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Solution
Solution Steps
Step 1: Find the velocity function \( v(t) \)
To find the acceleration function \( a(t) \), we first need to determine the velocity function \( v(t) \). The velocity is the derivative of the position function \( s(t) \).
Given:
\[
s(t) = 2t^{4} - 2t^{3} - 2t^{2} - 8
\]
Compute the derivative of \( s(t) \) with respect to \( t \):
\[
v(t) = \frac{ds}{dt} = \frac{d}{dt}(2t^{4}) - \frac{d}{dt}(2t^{3}) - \frac{d}{dt}(2t^{2}) - \frac{d}{dt}(8)
\]
Using the power rule for differentiation:
\[
v(t) = 8t^{3} - 6t^{2} - 4t
\]
Step 2: Find the acceleration function \( a(t) \)
The acceleration function \( a(t) \) is the derivative of the velocity function \( v(t) \).
Given:
\[
v(t) = 8t^{3} - 6t^{2} - 4t
\]
Compute the derivative of \( v(t) \) with respect to \( t \):
\[
a(t) = \frac{dv}{dt} = \frac{d}{dt}(8t^{3}) - \frac{d}{dt}(6t^{2}) - \frac{d}{dt}(4t)
\]
Using the power rule for differentiation:
\[
a(t) = 24t^{2} - 12t - 4
\]
Step 3: Write the final acceleration function
The acceleration function \( a(t) \) is:
\[
a(t) = 24t^{2} - 12t - 4
\]