An 8 kg block accelerates uniformly from rest to a velocity of `4 m s^(-1)` in 40 second. The instantaneous power at the end of 8 second is `P xx 10^(-2)` watt. Find the value of P.

Explanation

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

### Step 1: Determine the acceleration of the block.
The block accelerates uniformly from rest (initial velocity \( u = 0 \)) to a final velocity \( v = 4 \, \text{m/s} \) in a time \( t = 40 \, \text{s} \).

Using the formula for acceleration:
\[ a = \frac{v - u}{t} \]
Substituting the known values:
\[ a = \frac{4 \, \text{m/s} - 0 \, \text{m/s}}{40 \, \text{s}} = \frac{4}{40} = 0.1 \, \text{m/s}^2 \]

### Step 2: Calculate the instantaneous velocity after 8 seconds.
To find the instantaneous velocity at \( t = 8 \, \text{s} \), we can use the formula:
\[ v = u + a t \]
Substituting the values:
\[ v = 0 + (0.1 \, \text{m/s}^2)(8 \, \text{s}) = 0.8 \, \text{m/s} \]

### Step 3: Calculate the force acting on the block.
Using Newton's second law, the force \( F \) can be calculated as:
\[ F = m \cdot a \]
Where \( m = 8 \, \text{kg} \) and \( a = 0.1 \, \text{m/s}^2 \):
\[ F = 8 \, \text{kg} \cdot 0.1 \, \text{m/s}^2 = 0.8 \, \text{N} \]

### Step 4: Calculate the instantaneous power at 8 seconds.
The instantaneous power \( P \) can be calculated using the formula:
\[ P = F \cdot v \]
Substituting the values of force and instantaneous velocity:
\[ P = 0.8 \, \text{N} \cdot 0.8 \, \text{m/s} = 0.64 \, \text{W} \]

### Step 5: Express the power in the required format.
The problem states that the instantaneous power at the end of 8 seconds is \( P \times 10^{-2} \) watts. Thus:
\[ 0.64 \, \text{W} = P \times 10^{-2} \, \text{W} \]
To find \( P \):
\[ P = 0.64 \times 10^{2} = 64 \]

### Final Answer:
The value of \( P \) is \( 64 \).

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