The position of and 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=0`s and `t=2.0 s?` What is the average velocity between `t=2.0s` and `t=4.0 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 Position Function
The position of the object is given by the equation:
\[ x = a + bt^2 \]
where:
- \( a = 8.5 \, \text{m} \)
- \( b = 2.5 \, \text{m/s}^2 \)
- \( t \) is time in seconds.

### Step 2: Find the Velocity Function
Velocity is defined as the rate of change of position with respect to time. Therefore, we need to differentiate the position function with respect to time \( t \):
\[ v = \frac{dx}{dt} = \frac{d}{dt}(a + bt^2) \]
Since \( a \) is a constant, its derivative is zero. The derivative of \( bt^2 \) is:
\[ v = 0 + 2bt = 2bt \]
Substituting the value of \( b \):
\[ v = 2 \times 2.5 \times t = 5t \]

### Step 3: Calculate Velocity at \( t = 0 \, \text{s} \)
Now, we will find the velocity at \( t = 0 \):
\[ v(0) = 5 \times 0 = 0 \, \text{m/s} \]

### Step 4: Calculate Velocity at \( t = 2.0 \, \text{s} \)
Next, we will find the velocity at \( t = 2.0 \):
\[ v(2) = 5 \times 2 = 10 \, \text{m/s} \]

### Step 5: Calculate the Average Velocity between \( t = 2.0 \, \text{s} \) and \( t = 4.0 \, \text{s} \)
To find the average velocity, we need to calculate the displacement over the time interval. The formula for average velocity \( v_{avg} \) is:
\[ v_{avg} = \frac{x(t_2) - x(t_1)}{t_2 - t_1} \]
where \( t_1 = 2.0 \, \text{s} \) and \( t_2 = 4.0 \, \text{s} \).

### Step 6: Calculate Position at \( t = 4.0 \, \text{s} \)
Using the position function:
\[ x(4) = a + b(4^2) = 8.5 + 2.5 \times 16 = 8.5 + 40 = 48.5 \, \text{m} \]

### Step 7: Calculate Position at \( t = 2.0 \, \text{s} \)
Now calculate the position at \( t = 2.0 \):
\[ x(2) = a + b(2^2) = 8.5 + 2.5 \times 4 = 8.5 + 10 = 18.5 \, \text{m} \]

### Step 8: Substitute into the Average Velocity Formula
Now substitute the values into the average velocity formula:
\[ v_{avg} = \frac{x(4) - x(2)}{4 - 2} = \frac{48.5 - 18.5}{2} = \frac{30}{2} = 15 \, \text{m/s} \]

### Final Answers
- Velocity at \( t = 0 \, \text{s} \): \( 0 \, \text{m/s} \)
- Velocity at \( t = 2.0 \, \text{s} \): \( 10 \, \text{m/s} \)
- Average velocity between \( t = 2.0 \, \text{s} \) and \( t = 4.0 \, \text{s} \): \( 15 \, \text{m/s} \)

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