The equation of motion of a particle is s=4 - 5+³ +2²-t, where s is in meters and it is in seconds. (Assume t 2.0.) (a) Find the velocity, v(t), and acceleration, a(t), as functions of t. v(t) = 4³-15r²+2r a(t) = 121²-30r+2 +2r-1 (b) Find the acceleration (in m/s2) after 1.5 s. a(1.5) --16 ✓ m/s² (c) Graph the position, velocity, and acceleration functions on the same screen. V
The equation of motion of a particle is s=4 - 5+³ +2²-t, where s is in meters and it is in seconds. (Assume t 2.0.) (a) Find the velocity, v(t), and acceleration, a(t), as functions of t. v(t) = 4³-15r²+2r a(t) = 121²-30r+2 +2r-1 (b) Find the acceleration (in m/s2) after 1.5 s. a(1.5) --16 ✓ m/s² (c) Graph the position, velocity, and acceleration functions on the same screen. V
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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For the graph,
I believe the answer is the 3rd one but I'm also confused because the points for s(t) and a(t) are opposite, in terms of 5t3 in s(t) and 30t in a(t).
Can you explain if this is the answer and why it's correct?
![**The Equation of Motion of a Particle**
The equation of motion of a particle is given by \( s = t^4 - 5t^3 + 2t^2 - t \), where \( s \) is in meters and \( t \) is in seconds (assuming \( t \geq 0 \)).
**(a) Finding the Velocity and Acceleration Functions**
To find the velocity, \( v(t) \), and acceleration, \( a(t) \), functions of \( t \):
- The velocity function, \( v(t) \), is derived as:
\[
v(t) = 4t^3 - 15t^2 + 2t - 1
\]
- The acceleration function, \( a(t) \), is derived as:
\[
a(t) = 12t^2 - 30t + 2
\]
**(b) Finding the Acceleration at a Specific Time**
To find the acceleration in \( \text{m/s}^2 \) after \( 1.5 \) seconds:
- The acceleration at \( t = 1.5 \) is calculated as:
\[
a(1.5) = -16 \, \text{m/s}^2
\]
**(c) Graphing the Functions**
Graphs of the position, velocity, and acceleration functions are provided. Each graph displays the curves of the position (\( s \)), velocity (\( v \)), and acceleration (\( a \)) functions plotted over the same interval for a comprehensive visual comparison.
- The **position function** (in blue) shows the displacement of the particle over time.
- The **velocity function** (in red) depicts the rate of change of displacement, indicating how fast the particle’s position changes.
- The **acceleration function** (in purple) illustrates the rate of change of velocity, showing how the particle's speed changes over time.
Each graph is labeled with an \( x \)-axis (time \( t \), ranging from 0 to 5 seconds) and a \( y \)-axis (measurement units vary for different functions). The curves indicate how each variable changes as time progresses.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3eb03a80-7cae-4d33-b228-e2e0361c370e%2F1413d9a1-84f2-4d4a-8873-0179df935e1e%2F5iwvcu_processed.png&w=3840&q=75)
Transcribed Image Text:**The Equation of Motion of a Particle**
The equation of motion of a particle is given by \( s = t^4 - 5t^3 + 2t^2 - t \), where \( s \) is in meters and \( t \) is in seconds (assuming \( t \geq 0 \)).
**(a) Finding the Velocity and Acceleration Functions**
To find the velocity, \( v(t) \), and acceleration, \( a(t) \), functions of \( t \):
- The velocity function, \( v(t) \), is derived as:
\[
v(t) = 4t^3 - 15t^2 + 2t - 1
\]
- The acceleration function, \( a(t) \), is derived as:
\[
a(t) = 12t^2 - 30t + 2
\]
**(b) Finding the Acceleration at a Specific Time**
To find the acceleration in \( \text{m/s}^2 \) after \( 1.5 \) seconds:
- The acceleration at \( t = 1.5 \) is calculated as:
\[
a(1.5) = -16 \, \text{m/s}^2
\]
**(c) Graphing the Functions**
Graphs of the position, velocity, and acceleration functions are provided. Each graph displays the curves of the position (\( s \)), velocity (\( v \)), and acceleration (\( a \)) functions plotted over the same interval for a comprehensive visual comparison.
- The **position function** (in blue) shows the displacement of the particle over time.
- The **velocity function** (in red) depicts the rate of change of displacement, indicating how fast the particle’s position changes.
- The **acceleration function** (in purple) illustrates the rate of change of velocity, showing how the particle's speed changes over time.
Each graph is labeled with an \( x \)-axis (time \( t \), ranging from 0 to 5 seconds) and a \( y \)-axis (measurement units vary for different functions). The curves indicate how each variable changes as time progresses.
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