give the calculation step by step as like as full answer in 2 steps and explanation block in every steps.
And final answer summary
Definition of Parameters
Solutions are reviewed by our quality t

Question

give the calculation step by step as like as full answer in 2 steps and explanation block in every steps. 
And final answer summary
Definition of Parameters 
Solutions are reviewed by our quality team mainly based on 6 parameters: Relevancy, Completeness, Accuracy, Clarity, Structure, and Voice. 
Relevancy: This indicates how closely a solution aligns with the question asked. It evaluates whether the solution addresses the core aspects of the question and provides appropriate information. 
Completeness: This indicates how thoroughly the solution addresses various asks in the question. It evaluates whether the solution has all essential information like calculations, diagrams, procedural steps, & explanatory statements in the solution. 
Accuracy: It refers to how correct the provided solution is for the given question/s. This evaluates the correctness of the information/concept/method followed in the solution. 
Clarity: Clarity refers to how easily the solution can be understood by the student. This evaluates the language complexity, readability, and redundancy of a solution. 
Structure: Structure refers to how the solution is organized and presented in relation to the requi

VIPSolutions · Accepted Answer

**Question:**  
A particle follows the trajectory  
$ x(t) = \frac{1}{4}t^3 + 2t $  
$ y(t) = 20t - t^2 $

---

### Step 1: Find the Velocity Components

**Explanation:**  
The velocity of the particle in each direction is found by differentiating the position functions with respect to time $ t $.

#### Calculations:
$$
v_x(t) = \frac{dx}{dt} = \frac{d}{dt}\left(\frac{1}{4}t^3 + 2t\right)
$$
$$
= \frac{3}{4}t^2 + 2
$$

$$
v_y(t) = \frac{dy}{dt} = \frac{d}{dt}(20t - t^2)
$$
$$
= 20 - 2t
$$

---

### Step 2: Find the Acceleration Components

**Explanation:**  
Acceleration is found by differentiating the velocity components with respect to time $ t $.

#### Calculations:
$$
a_x(t) = \frac{dv_x}{dt} = \frac{d}{dt}\left(\frac{3}{4}t^2 + 2\right)
$$
$$
= \frac{3}{2}t
$$

$$
a_y(t) = \frac{dv_y}{dt} = \frac{d}{dt}(20 - 2t)
$$
$$
= -2
$$

---

### Final Answer Summary

**Velocity components:**  
$ v_x(t) = \frac{3}{4}t^2 + 2 $  
$ v_y(t) = 20 - 2t $

**Acceleration components:**  
$ a_x(t) = \frac{3}{2}t $  
$ a_y(t) = -2 $

---

#### Definition of Parameters Applied

- **Relevancy:** The calculations directly address the question about the motion of the particle.
- **Completeness:** All necessary steps (finding velocity and acceleration) are provided and explained.
- **Accuracy:** Correct differentiation is used for both components.
- **Clarity:** Each step is labeled and briefly explained for easy understanding.
- **Structure:** The solution progresses logically from position to velocity to acceleration.
- **Voice:** The explanation uses a clear and instructional tone.

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