2. a. A particle is undergoing a uniform rectilinear motion and the displacement as a function of time is given by r(t) = 6tª- 2t³ -12t²+3t+3 where r is in meters and t is in seconds. b. What is a uniform rectilinear motion? c. Find the average acceleration of the particle between t = 0 and t=2s d. When is the acceleration zero?
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.
![**Physics - Motion of Particles**
**2.**
**a.** A particle is undergoing a uniform rectilinear motion and the displacement as a function of time is given by:
\[
\mathbf{r}(t) = 6t^4 - 2t^3 - 12t^2 + 3t + 3
\]
where \( r \) is in meters and \( t \) is in seconds.
**b.** What is a uniform rectilinear motion?
**c.** Find the average acceleration of the particle between \( t = 0 \) and \( t = 2s \).
**d.** When is the acceleration zero?
**3.** The position of a particular particle as a function of time is given by:
\[
\mathbf{r} = (9.60 t \, \mathbf{i} + 8.58 \, \mathbf{j} - 1.00 t^2 \, \mathbf{k}) \text{ meters}
\]
Determine the particle’s velocity and acceleration as a function of time.
---
### Detailed Explanation:
**2.**
**(a)** The given function describes the displacement \(\mathbf{r}(t)\) of a particle in uniform rectilinear motion. The displacement is represented by a polynomial equation in terms of time \(t\).
**(b)** Uniform rectilinear motion refers to motion in a straight line with uniform (constant) velocity, meaning that both the direction and speed of the motion remain consistent over time.
**(c)** To find the average acceleration, you need to calculate the change in velocity over the change in time. The first step is to differentiate the displacement function \(\mathbf{r}(t)\) to find the velocity function \( \mathbf{v}(t) \):
\[
\mathbf{v}(t) = \frac{d}{dt}[6t^4 - 2t^3 - 12t^2 + 3t + 3]
\]
Then, differentiate the velocity function to find the acceleration function \( \mathbf{a}(t) \):
\[
\mathbf{a}(t) = \frac{d}{dt} \mathbf{v}(t)
\]
Evaluate \( \mathbf{a}(t) \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F64a30517-e863-4662-9804-11d4230920b7%2Fb8ab5c8b-f0f5-4d28-b635-74f7e5dc5aa7%2F50vitf_processed.jpeg&w=3840&q=75)
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