A block of mass `m` at rest is acted upon by a force `F` for a time t. The kinetic energy of block after time t is

Options

A. `(F^(2)t^(2))/(m)`
B. `(F^(2)t^(2))/(2m)`
C. `(F^(2)t^(2)/(3m))`
D. `(Ft)/(2m)`

Explanation

To find the kinetic energy of a block of mass \( m \) after being acted upon by a force \( F \) for a time \( t \), we can follow these steps:

### Step 1: Understand the relationship between force, mass, and acceleration
According to Newton's second law, the force \( F \) acting on an object is equal to the mass \( m \) of the object multiplied by its acceleration \( a \):
\[ F = m \cdot a \]
From this, we can express acceleration as:
\[ a = \frac{F}{m} \]

### Step 2: Relate acceleration to velocity
Since the block starts from rest, we can use the kinematic equation that relates acceleration to change in velocity over time:
\[ v = u + at \]
Where \( u \) is the initial velocity (which is 0 since the block is at rest), thus:
\[ v = 0 + \left(\frac{F}{m}\right) t = \frac{F t}{m} \]

### Step 3: Calculate the momentum
The momentum \( p \) of the block can be calculated using the formula:
\[ p = m \cdot v \]
Substituting the expression for \( v \):
\[ p = m \cdot \left(\frac{F t}{m}\right) = F t \]

### Step 4: Use the momentum to find kinetic energy
The kinetic energy \( KE \) of the block is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
Substituting \( v \) from Step 2:
\[ KE = \frac{1}{2} m \left(\frac{F t}{m}\right)^2 \]
This simplifies to:
\[ KE = \frac{1}{2} m \cdot \frac{F^2 t^2}{m^2} = \frac{F^2 t^2}{2m} \]

### Final Answer
Thus, the kinetic energy of the block after time \( t \) is:
\[ KE = \frac{F^2 t^2}{2m} \]