body of mass 10 covers 20 m in its 2nd second and 50 m in its 3rd second of its journey. 
Compute its initial velocity and acceleration.

To solve this problem, we'll use the equations of motion for uniformly accelerated linear motion. Let's define the variables first:

- \( u \): initial velocity (m/s)
- \( a \): acceleration (m/s²)
- \( s_n \): distance covered in the \( n \)th second
- \( t_n \): time at the \( n \)th second

Given:
- Mass of the body, \( m = 10 \) kg (though not needed to solve this problem)
- Distance covered in the 2nd second, \( s_2 = 20 \) m
- Distance covered in the 3rd second, \( s_3 = 50 \) m

### Equations to use:
The distance covered in the \( n \)th second is given by:
\[
s_n = u + \frac{a}{2} \cdot (2n - 1)
\]

### Step 1: Using the equation for the 2nd second

For \( n = 2 \):
\[
s_2 = u + \frac{a}{2} \cdot (2 \times 2 - 1) = u + \frac{3a}{2}
\]
Given \( s_2 = 20 \) m, we have:
\[
20 = u + \frac{3a}{2} \quad \text{(Equation 1)}
\]

### Step 2: Using the equation for the 3rd second

For \( n = 3 \):
\[
s_3 = u + \frac{a}{2} \cdot (2 \times 3 - 1) = u + \frac{5a}{2}
\]
Given \( s_3 = 50 \) m, we have:
\[
50 = u + \frac{5a}{2} \quad \text{(Equation 2)}
\]

### Step 3: Solving the equations

Subtract Equation 1 from Equation 2:
\[
50 - 20 = \left(u + \frac{5a}{2}\right) - \left(u + \frac{3a}{2}\right)
\]
\[
30 = \frac{5a}{2} - \frac{3a}{2} = \frac{2a}{2} = a
\]

So, \( a = 30 \) m/s².

Now, substitute \( a = 30 \) m/s² back into Equation 1:
\[
20 = u + \frac{3 \times 30}{2}
\]
\[
20 = u + 45
\]
\[
u = 20 - 45 = -25 \text{ m/s}
\]

### Final Answers:
- **Initial velocity \( u = -25 \) m/s** (the negative sign indicates the direction is opposite to the chosen positive direction).
- **Acceleration \( a = 30 \) m/s²**.

Would you like further details on any part of this solution or have any other questions?

### Related Questions:
1. How would the time of travel change if the acceleration was halved?
2. What would be the final velocity after 5 seconds of motion?
3. How does the mass of the body affect the acceleration and velocity in this context?
4. What would be the displacement after the first 3 seconds?
5. How would the problem change if there was a constant opposing force?

### Tip:
Always check the direction of motion when working with initial velocities and accelerations, especially if the initial velocity is negative, as it can affect the interpretation of the results.

Learn how to calculate the initial velocity and acceleration of a body in uniformly accelerated motion using equations of motion. Given specific distances covered in the 2nd and 3rd seconds, determine the initial conditions of motion. Suitable for high school physics students.

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