A body is moving with uniform acceleration covers 200 m in the first 2 s and 220 m in the next 4s. Find the velocity in `ms ^(-1) ` after 7 s.

Options

A. 10
B. 15
C. 20
D. 30

Answer

To solve the problem step by step, we will use the equations of motion under uniform acceleration.

### Step 1: Analyze the first part of the motion
We know that the body covers 200 m in the first 2 seconds. We can use the second equation of motion:
\[ S = ut + \frac{1}{2} a t^2 \]
Where:
- \( S \) = distance covered (200 m)
- \( u \) = initial velocity (unknown)
- \( a \) = acceleration (unknown)
- \( t \) = time (2 s)

Substituting the values into the equation:
\[ 200 = 2u + \frac{1}{2} a (2^2) \]
\[ 200 = 2u + 2a \]
Dividing the entire equation by 2:
\[ 100 = u + a \]
This is our **Equation 1**.

### Step 2: Analyze the second part of the motion
The body covers an additional 220 m in the next 4 seconds (from t = 2 s to t = 6 s). The total distance covered in 6 seconds is:
\[ S = 200 + 220 = 420 \, \text{m} \]
Now we can use the same equation of motion:
\[ S = ut + \frac{1}{2} a t^2 \]
Where:
- \( S = 420 \, \text{m} \)
- \( t = 6 \, \text{s} \)

Substituting the values:
\[ 420 = 6u + \frac{1}{2} a (6^2) \]
\[ 420 = 6u + 18a \]
Dividing the entire equation by 6:
\[ 70 = u + 3a \]
This is our **Equation 2**.

### Step 3: Solve the equations simultaneously
Now we have two equations:
1. \( 100 = u + a \) (Equation 1)
2. \( 70 = u + 3a \) (Equation 2)

To eliminate \( u \), we can subtract Equation 2 from Equation 1:
\[ (100 - 70) = (u + a) - (u + 3a) \]
\[ 30 = -2a \]
Thus, we find:
\[ a = -15 \, \text{m/s}^2 \]

### Step 4: Substitute \( a \) back to find \( u \)
Now substitute \( a \) back into Equation 1:
\[ 100 = u + (-15) \]
\[ u = 100 + 15 \]
\[ u = 115 \, \text{m/s} \]

### Step 5: Calculate the velocity after 7 seconds
Now we can find the velocity after 7 seconds using the first equation of motion:
\[ v = u + at \]
Where:
- \( u = 115 \, \text{m/s} \)
- \( a = -15 \, \text{m/s}^2 \)
- \( t = 7 \, \text{s} \)

Substituting the values:
\[ v = 115 + (-15)(7) \]
\[ v = 115 - 105 \]
\[ v = 10 \, \text{m/s} \]

### Final Answer
The velocity of the body after 7 seconds is **10 m/s**.

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