O 8. The acceleration a of an object moving along a line is given by the equation a(t) = -32, with s(0) = 0 and s'(0) = 10. What is a formula for the position function s(t)?
O 8. The acceleration a of an object moving along a line is given by the equation a(t) = -32, with s(0) = 0 and s'(0) = 10. What is a formula for the position function s(t)?
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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![**Problem Statement:**
The acceleration \( a \) of an object moving along a line is given by the equation \( a(t) = -32 \), with initial conditions \( s(0) = 0 \) and \( s'(0) = 10 \). What is a formula for the position function \( s(t) \)?
**Solution Approach:**
1. **Understanding Acceleration and Velocity:**
- The acceleration function \( a(t) = -32 \) indicates constant acceleration.
- The relationship between acceleration and velocity is given by the derivative: \( v(t) = \int a(t) \, dt \).
2. **Calculating the Velocity Function:**
- Integrate \( a(t) = -32 \) with respect to \( t \):
\[
v(t) = \int -32 \, dt = -32t + C
\]
- Use the initial condition \( s'(0) = 10 \) (where \( s'(t) = v(t) \)):
\[
v(0) = -32(0) + C = 10 \implies C = 10
\]
- Thus, the velocity function is \( v(t) = -32t + 10 \).
3. **Determining the Position Function:**
- The relationship between velocity and position is given by the derivative: \( s(t) = \int v(t) \, dt \).
- Integrate \( v(t) = -32t + 10 \) with respect to \( t \):
\[
s(t) = \int (-32t + 10) \, dt = -16t^2 + 10t + D
\]
- Use the initial condition \( s(0) = 0 \):
\[
s(0) = -16(0)^2 + 10(0) + D = 0 \implies D = 0
\]
- Therefore, the position function is \( s(t) = -16t^2 + 10t \).
**Conclusion:**
The formula for the position function is \( s(t) = -16t^2 + 10t \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e3da8d7-5e45-4b35-92f7-ecdb5bd91ee7%2Fe43262ae-e740-4551-ab33-d1802d5d49f0%2Fud3gt0p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
The acceleration \( a \) of an object moving along a line is given by the equation \( a(t) = -32 \), with initial conditions \( s(0) = 0 \) and \( s'(0) = 10 \). What is a formula for the position function \( s(t) \)?
**Solution Approach:**
1. **Understanding Acceleration and Velocity:**
- The acceleration function \( a(t) = -32 \) indicates constant acceleration.
- The relationship between acceleration and velocity is given by the derivative: \( v(t) = \int a(t) \, dt \).
2. **Calculating the Velocity Function:**
- Integrate \( a(t) = -32 \) with respect to \( t \):
\[
v(t) = \int -32 \, dt = -32t + C
\]
- Use the initial condition \( s'(0) = 10 \) (where \( s'(t) = v(t) \)):
\[
v(0) = -32(0) + C = 10 \implies C = 10
\]
- Thus, the velocity function is \( v(t) = -32t + 10 \).
3. **Determining the Position Function:**
- The relationship between velocity and position is given by the derivative: \( s(t) = \int v(t) \, dt \).
- Integrate \( v(t) = -32t + 10 \) with respect to \( t \):
\[
s(t) = \int (-32t + 10) \, dt = -16t^2 + 10t + D
\]
- Use the initial condition \( s(0) = 0 \):
\[
s(0) = -16(0)^2 + 10(0) + D = 0 \implies D = 0
\]
- Therefore, the position function is \( s(t) = -16t^2 + 10t \).
**Conclusion:**
The formula for the position function is \( s(t) = -16t^2 + 10t \).
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