Formulae And Concepts For Uniformaly Accelerated Motion In A Straight

Options

A. `u=r+at`
B. `s=((u+v)/(2))t`
C. `s=vt-(1)/(2)at^(2)`
D. `s=(v^(2)-u^(2))/(2a)`

Answer

To solve the question regarding the formulae and concepts for uniformly accelerated motion in a straight line, we will identify the correct equations of motion. Here’s a step-by-step breakdown:

### Step 1: Understand the Concept of Uniformly Accelerated Motion
Uniformly accelerated motion refers to motion in which the acceleration is constant. This means that the rate of change of velocity is the same over time.

### Step 2: Recall the Three Equations of Motion
The three fundamental equations of motion for uniformly accelerated motion are:
1. \( v = u + at \)
2. \( s = ut + \frac{1}{2} a t^2 \)
3. \( v^2 = u^2 + 2as \)

Where:
- \( v \) = final velocity
- \( u \) = initial velocity
- \( a \) = acceleration
- \( s \) = displacement
- \( t \) = time

### Step 3: Analyze Each Given Equation
Now, we will analyze the four equations provided in the question to determine which one is correct.

1. **Equation 1**: \( u = r + at \)
- This equation does not match any of the three equations of motion. The variable \( r \) is not defined in the context of uniformly accelerated motion.

2. **Equation 2**: \( s = \frac{u + v}{2} \cdot t \)
- This equation is a rearrangement of the second equation of motion. It is valid and represents the average velocity multiplied by time to give displacement.

3. **Equation 3**: \( s = vt - \frac{1}{2} at^2 \)
- This equation has a negative sign and uses \( v \) instead of \( u \). It does not match the second equation of motion, hence it is incorrect.

4. **Equation 4**: \( s = \frac{v^2 - u^2}{2a} \)
- This can be rearranged to \( v^2 = u^2 + 2as \), which matches the third equation of motion. Therefore, this equation is valid.

### Step 4: Conclusion
From the analysis, the correct equations for uniformly accelerated motion are:
- The second equation \( s = \frac{u + v}{2} \cdot t \) is valid.
- The fourth equation \( s = \frac{v^2 - u^2}{2a} \) is also valid as it can be rearranged to match one of the standard equations of motion.

### Final Answer
The correct options for uniformly accelerated motion are:
- \( s = \frac{u + v}{2} \cdot t \) (Equation 2)
- \( s = \frac{v^2 - u^2}{2a} \) (Equation 4)