Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![### Problem:
Given an acceleration function \(a(t) = \langle 0, -32 \rangle\), find the velocity and position functions, given that
\[ v(0) = \langle 10, 50 \rangle \quad \text{and} \quad r(0) = \langle 0, 0 \rangle. \]
### Solution:
This problem involves finding the velocity and position vectors from the given acceleration vector. Here’s the step-by-step process to solve it:
#### Step 1: Find the Velocity Function
The acceleration \(a(t) = \langle 0, -32 \rangle\) gives the rate of change of velocity.
\[ v(t) = \int a(t) \, dt \]
Integrate each component of \(a(t)\):
\[
a(t) = \langle 0, -32 \rangle
\]
\[
\int 0 \, dt = C_1 \quad \text{and} \quad \int -32 \, dt = -32t + C_2
\]
So,
\[
v(t) = \langle C_1, -32t + C_2 \rangle
\]
Using the given initial condition \(v(0) = \langle 10, 50 \rangle\), we can find \(C_1\) and \(C_2\):
\[
v(0) = \langle C_1, C_2 \rangle = \langle 10, 50 \rangle
\]
Thus,
\[
C_1 = 10 \quad \text{and} \quad C_2 = 50
\]
So, the velocity function is:
\[
v(t) = \langle 10, -32t + 50 \rangle
\]
#### Step 2: Find the Position Function
The velocity \(v(t) = \langle 10, -32t + 50 \rangle\) gives the rate of change of position.
\[
r(t) = \int v(t) \, dt
\]
Integrate each component of \(v(t)\):
\[
\int 10 \, dt = 10t + D_1 \quad \text{and} \quad \int (-32t + 50) \,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8613cfca-9b11-4abe-bfa6-81adbc09aa49%2Fd6c73e36-2096-4c36-86fc-99e6afe29848%2F5t0w3n9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem:
Given an acceleration function \(a(t) = \langle 0, -32 \rangle\), find the velocity and position functions, given that
\[ v(0) = \langle 10, 50 \rangle \quad \text{and} \quad r(0) = \langle 0, 0 \rangle. \]
### Solution:
This problem involves finding the velocity and position vectors from the given acceleration vector. Here’s the step-by-step process to solve it:
#### Step 1: Find the Velocity Function
The acceleration \(a(t) = \langle 0, -32 \rangle\) gives the rate of change of velocity.
\[ v(t) = \int a(t) \, dt \]
Integrate each component of \(a(t)\):
\[
a(t) = \langle 0, -32 \rangle
\]
\[
\int 0 \, dt = C_1 \quad \text{and} \quad \int -32 \, dt = -32t + C_2
\]
So,
\[
v(t) = \langle C_1, -32t + C_2 \rangle
\]
Using the given initial condition \(v(0) = \langle 10, 50 \rangle\), we can find \(C_1\) and \(C_2\):
\[
v(0) = \langle C_1, C_2 \rangle = \langle 10, 50 \rangle
\]
Thus,
\[
C_1 = 10 \quad \text{and} \quad C_2 = 50
\]
So, the velocity function is:
\[
v(t) = \langle 10, -32t + 50 \rangle
\]
#### Step 2: Find the Position Function
The velocity \(v(t) = \langle 10, -32t + 50 \rangle\) gives the rate of change of position.
\[
r(t) = \int v(t) \, dt
\]
Integrate each component of \(v(t)\):
\[
\int 10 \, dt = 10t + D_1 \quad \text{and} \quad \int (-32t + 50) \,
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