**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 \( \mathbf{a}(t) \)

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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?

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?

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Concept explainers

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.

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Question

**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 \( \mathbf{a}(t) \)

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