The position of an object moving along x-axis is given by `x=a +bt^(2)`, where `a=8.5 m` and b=2.5 ms^(-2) and (t) is measured in seconds. What is the velocity at t=1= 0`s and `t=2.0 s? What is the average velocity between `t=2.0s` and `t=4.0 s`? a) 5 m/s b) 10 m/s c) 15 m/s d) 20 m/s

Options

A. `5 m s^(-1)`
B. `10 m s^(-1)`
C. ` 15 m s^(-1)`
D. `20 m s^(-1)`

Explanation

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the given equation
The position of the object is given by the equation:
\[ x = a + bt^2 \]
where \( a = 8.5 \, \text{m} \) and \( b = 2.5 \, \text{m/s}^2 \).

### Step 2: Find the velocity function
Velocity \( v \) is the derivative of position \( x \) with respect to time \( t \):
\[ v = \frac{dx}{dt} \]
To find this, we differentiate the position equation:
\[ v = \frac{d}{dt}(a + bt^2) = 0 + 2bt = 2bt \]

### Step 3: Calculate the velocity at \( t = 0 \, \text{s} \)
Substituting \( t = 0 \) into the velocity equation:
\[ v(0) = 2b(0) = 0 \, \text{m/s} \]

### Step 4: Calculate the velocity at \( t = 1 \, \text{s} \)
Substituting \( t = 1 \) into the velocity equation:
\[ v(1) = 2b(1) = 2 \times 2.5 = 5 \, \text{m/s} \]

### Step 5: Calculate the velocity at \( t = 2 \, \text{s} \)
Substituting \( t = 2 \) into the velocity equation:
\[ v(2) = 2b(2) = 2 \times 2.5 \times 2 = 10 \, \text{m/s} \]

### Step 6: Calculate the average velocity between \( t = 2 \, \text{s} \) and \( t = 4 \, \text{s} \)
First, we need to find the positions at \( t = 2 \) and \( t = 4 \).

- **Position at \( t = 2 \, \text{s} \)**:
\[ x(2) = a + b(2^2) = 8.5 + 2.5 \times 4 = 8.5 + 10 = 18.5 \, \text{m} \]

- **Position at \( t = 4 \, \text{s} \)**:
\[ x(4) = a + b(4^2) = 8.5 + 2.5 \times 16 = 8.5 + 40 = 48.5 \, \text{m} \]

### Step 7: Calculate the total displacement
The total displacement from \( t = 2 \) to \( t = 4 \) is:
\[ \Delta x = x(4) - x(2) = 48.5 - 18.5 = 30 \, \text{m} \]

### Step 8: Calculate the time interval
The time interval is:
\[ \Delta t = 4 - 2 = 2 \, \text{s} \]

### Step 9: Calculate the average velocity
The average velocity \( v_{avg} \) is given by:
\[ v_{avg} = \frac{\Delta x}{\Delta t} = \frac{30 \, \text{m}}{2 \, \text{s}} = 15 \, \text{m/s} \]

### Final Answer
The average velocity between \( t = 2.0 \, \text{s} \) and \( t = 4.0 \, \text{s} \) is:
\[ \boxed{15 \, \text{m/s}} \]