At the start of a motion along a line the initial velocity is u and acceleration is at. The final velocity vafter time t, is

Options

A. `v= u + at`
B. `v = u + at^(2)`
C. `v = u +(1)/(2) at^(2)`
D. `v = at^(2)`

Explanation

To find the final velocity \( v \) after time \( t \) given the initial velocity \( u \) and acceleration \( a \), we can follow these steps:

### Step 1: Understand the relationship between acceleration, velocity, and time.
Acceleration \( a \) is defined as the rate of change of velocity with respect to time. Mathematically, this is expressed as:
\[ a = \frac{dv}{dt} \]

### Step 2: Rearrange the equation for integration.
To find the change in velocity, we can rearrange the equation:
\[ dv = a \, dt \]

### Step 3: Integrate both sides.
We will integrate both sides to find the relationship between velocity and time. The left-hand side will be integrated with respect to \( v \) and the right-hand side with respect to \( t \):
\[ \int dv = \int a \, dt \]

### Step 4: Set the limits of integration.
The limits for the left-hand side (velocity) will be from the initial velocity \( u \) to the final velocity \( v \), and for the right-hand side (time), from \( 0 \) to \( t \):
\[ \int_{u}^{v} dv = \int_{0}^{t} a \, dt \]

### Step 5: Solve the integrals.
The left-hand side gives:
\[ v - u \]
The right-hand side, assuming \( a \) is constant, gives:
\[ at \]

### Step 6: Combine the results.
Now we can combine the results from both sides:
\[ v - u = at \]

### Step 7: Solve for final velocity \( v \).
Rearranging the equation gives us:
\[ v = u + at \]

### Conclusion:
The final velocity \( v \) after time \( t \) is:
\[ v = u + at \]