The velocity of a moving particle is given by the equation `v = (5 - t^(2))` The average acceleration of the particle between the 2nd and 3rd seconds is

Options

A. `-5ms^(-2)`
B. `2ma^(-2)`
C. `4ms^(-2)`
D. `-4ms^(-2)`

Explanation

To find the average acceleration of the particle between the 2nd and 3rd seconds, we can follow these steps:

### Step 1: Understand the Formula for Average Acceleration
Average acceleration (\(a_{avg}\)) is defined as the change in velocity (\(\Delta v\)) divided by the change in time (\(\Delta t\)):
\[ a_{avg} = \frac{\Delta v}{\Delta t} \]
where \(\Delta v = v_f - v_i\) (final velocity - initial velocity) and \(\Delta t = t_f - t_i\) (final time - initial time).

### Step 2: Identify the Time Interval
In this case, we need to calculate the average acceleration between the 2nd second (\(t_i = 2\) seconds) and the 3rd second (\(t_f = 3\) seconds).

### Step 3: Calculate the Change in Time
\[ \Delta t = t_f - t_i = 3 - 2 = 1 \text{ second} \]

### Step 4: Calculate the Initial Velocity (\(v_i\)) at \(t = 2\) seconds
Using the given velocity equation \(v = 5 - t^2\):
\[ v_i = v(2) = 5 - (2^2) = 5 - 4 = 1 \text{ m/s} \]

### Step 5: Calculate the Final Velocity (\(v_f\)) at \(t = 3\) seconds
Using the same velocity equation:
\[ v_f = v(3) = 5 - (3^2) = 5 - 9 = -4 \text{ m/s} \]

### Step 6: Calculate the Change in Velocity
\[ \Delta v = v_f - v_i = -4 - 1 = -5 \text{ m/s} \]

### Step 7: Substitute Values into the Average Acceleration Formula
Now we can substitute the values of \(\Delta v\) and \(\Delta t\) into the average acceleration formula:
\[ a_{avg} = \frac{\Delta v}{\Delta t} = \frac{-5 \text{ m/s}}{1 \text{ s}} = -5 \text{ m/s}^2 \]

### Final Answer
The average acceleration of the particle between the 2nd and 3rd seconds is \(-5 \text{ m/s}^2\).

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