A body travels a distance of 2 m in 2 seconds and 2.2m next 4 secs. What will be the velocity of the body at the end of `7 th second from the start ?

Options

A. ` 5 cms^(-1)`
B. ` 10 cm s^(-1)`
C. ` 15 cm s^(-1)`
D. ` 20 cm s^(-1)`

Explanation

To solve the problem step-by-step, we will break down the motion of the body into two parts based on the given distances and times.

### Step 1: Analyze the first part of the motion
The body travels a distance of 2 meters in 2 seconds.

- **Distance (s1)** = 2 m
- **Time (t1)** = 2 s

Using the equation of motion:
\[ s = ut + \frac{1}{2}at^2 \]
where:
- \( s \) = distance traveled
- \( u \) = initial velocity
- \( a \) = acceleration
- \( t \) = time

For the first part:
\[ 2 = 2u + \frac{1}{2}a(2^2) \]
\[ 2 = 2u + 2a \]
Dividing the entire equation by 2:
\[ 1 = u + a \]
This is our **Equation 1**.

### Step 2: Analyze the second part of the motion
The body travels 2.2 meters in the next 4 seconds (from t = 2s to t = 6s).

- **Distance (s2)** = 2.2 m
- **Time (t2)** = 4 s
- **Total time at this point** = 6 s
- **Total distance traveled** = 2 m + 2.2 m = 4.2 m

Using the same equation of motion:
\[ 4.2 = 6u + \frac{1}{2}a(6^2) \]
\[ 4.2 = 6u + 18a \]
Dividing the entire equation by 6:
\[ 0.7 = u + 3a \]
This is our **Equation 2**.

### Step 3: Solve the equations
Now we have two equations:
1. \( 1 = u + a \) (Equation 1)
2. \( 0.7 = u + 3a \) (Equation 2)

Subtract Equation 1 from Equation 2:
\[ (0.7 - 1) = (u + 3a) - (u + a) \]
\[ -0.3 = 2a \]
Thus,
\[ a = -0.15 \, \text{m/s}^2 \]

### Step 4: Find the initial velocity
Substituting the value of \( a \) back into Equation 1:
\[ 1 = u - 0.15 \]
\[ u = 1 + 0.15 = 1.15 \, \text{m/s} \]

### Step 5: Calculate the final velocity at the end of the 7th second
To find the velocity at the end of the 7th second, we use the formula:
\[ v = u + at \]
where \( t = 7 \) seconds.
Substituting the values:
\[ v = 1.15 + (-0.15) \times 7 \]
\[ v = 1.15 - 1.05 \]
\[ v = 0.1 \, \text{m/s} \]

### Step 6: Convert to centimeters per second
Since the final answer needs to be in centimeters per second:
\[ 0.1 \, \text{m/s} = 0.1 \times 100 \, \text{cm/s} = 10 \, \text{cm/s} \]

### Final Answer
The velocity of the body at the end of the 7th second is **10 cm/s**.

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