In the s-t equation `(s=10+20 t-5t^(2))` match the following columns.
`|{:(,"Column I",,"Column II"),((A),"Distancec travelled in 3s",(p),-20" units"),((B),"Displacement 1 s",(q),15" units"),((C ),"Initial acceleration",(r ),25" units"),((D),"Velocity at 4 s",(s),-10" units"):}|`

Explanation

To solve the problem, we need to analyze the given equation of motion and find the required quantities step by step.

### Given:
The equation of motion is:
\[ s = 10 + 20t - 5t^2 \]

### Step 1: Calculate Distance Travelled in 3 seconds (A)
To find the distance travelled in 3 seconds, we substitute \( t = 3 \) into the equation:
\[ s = 10 + 20(3) - 5(3^2) \]
Calculating this step-by-step:
- \( 20(3) = 60 \)
- \( 5(3^2) = 5(9) = 45 \)
- Now substitute these values:
\[ s = 10 + 60 - 45 = 25 \text{ units} \]
Thus, the distance travelled in 3 seconds is **25 units**.

### Step 2: Calculate Displacement in 1 second (B)
Next, we find the displacement at \( t = 1 \):
\[ s = 10 + 20(1) - 5(1^2) \]
Calculating this:
- \( 20(1) = 20 \)
- \( 5(1^2) = 5(1) = 5 \)
- Now substitute:
\[ s = 10 + 20 - 5 = 25 \text{ units} \]
Thus, the displacement in 1 second is also **25 units**.

### Step 3: Find Initial Acceleration (C)
The acceleration can be found by first determining the velocity \( v \) from the displacement equation:
\[ v = \frac{ds}{dt} = \frac{d}{dt}(10 + 20t - 5t^2) \]
Differentiating:
- The derivative of \( 10 \) is \( 0 \)
- The derivative of \( 20t \) is \( 20 \)
- The derivative of \( -5t^2 \) is \( -10t \)

Thus, the velocity equation is:
\[ v = 20 - 10t \]
Now, to find the acceleration \( a \):
\[ a = \frac{dv}{dt} = \frac{d}{dt}(20 - 10t) \]
Differentiating:
- The derivative of \( 20 \) is \( 0 \)
- The derivative of \( -10t \) is \( -10 \)

Thus, the acceleration is:
\[ a = -10 \text{ units} \]

### Step 4: Calculate Velocity at 4 seconds (D)
Now, we find the velocity at \( t = 4 \):
\[ v = 20 - 10(4) \]
Calculating:
\[ v = 20 - 40 = -20 \text{ units} \]

### Summary of Results:
- A: Distance travelled in 3s = **25 units** (matches with R)
- B: Displacement in 1s = **25 units** (matches with R)
- C: Initial acceleration = **-10 units** (matches with S)
- D: Velocity at 4s = **-20 units** (matches with P)

### Final Matching:
- A matches with R
- B matches with R
- C matches with S
- D matches with P