Given the position function x(t), calculate the velocity function v(t) and the acceleration function a(t). a. x(e)=(3 m +| 2 +7m 3s-1 b. x(1) =(2m)e
Displacement, Velocity and Acceleration
In classical mechanics, kinematics deals with the motion of a particle. It deals only with the position, velocity, acceleration, and displacement of a particle. It has no concern about the source of motion.
Linear Displacement
The term "displacement" refers to when something shifts away from its original "location," and "linear" refers to a straight line. As a result, “Linear Displacement” can be described as the movement of an object in a straight line along a single axis, for example, from side to side or up and down. Non-contact sensors such as LVDTs and other linear location sensors can calculate linear displacement. Non-contact sensors such as LVDTs and other linear location sensors can calculate linear displacement. Linear displacement is usually measured in millimeters or inches and may be positive or negative.
![### Calculating Velocity and Acceleration from Position Functions
**Problem Statement:**
Given the position function \( x(t) \), calculate the velocity function \( v(t) \) and the acceleration function \( a(t) \).
**Solution:**
### Part (a)
Given:
\[ x(t) = \left(\frac{3 \, \text{m}}{ \text{s}^4}\right)t^4 + \left(\frac{2 \, \text{m}}{\text{s}^2}\right)t^2 + 7 \, \text{m} \]
1. **Velocity Function \( v(t) \)**:
The velocity function \( v(t) \) is the first derivative of the position function \( x(t) \) with respect to time \( t \):
\[
v(t) = \frac{d}{dt} x(t)
\]
Calculation:
\[
v(t) = \frac{d}{dt} \left[ \left(\frac{3 \, \text{m}}{ \text{s}^4}\right)t^4 + \left(\frac{2 \, \text{m}}{\text{s}^2}\right)t^2 + 7 \, \text{m} \right]
\]
\[
v(t) = 4 \left(\frac{3 \, \text{m}}{\text{s}^4}\right)t^3 + 2 \left(\frac{2 \, \text{m}}{\text{s}^2}\right) t
\]
\[
v(t) = \left(\frac{12 \, \text{m}}{\text{s}^4}\right)t^3 + \left(\frac{4 \, \text{m}}{\text{s}^2}\right)t
\]
2. **Acceleration Function \( a(t) \)**:
The acceleration function \( a(t) \) is the first derivative of the velocity function \( v(t) \) with respect to time \( t \):
\[
a(t) = \frac{d}{dt} v(t)
\]
Calculation:
\[
a(t) = \frac{d}{dt} \left[ \left(\frac{12 \, \text{m}}{\text](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdedc6ff1-e5e7-48e1-8e69-1d621b5c4e4e%2Fdd1f4c51-4187-40c1-a5c5-6932f2f3e0fd%2F5mkx44b_processed.jpeg&w=3840&q=75)
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