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.