A football is rolling down a hill of unknown shape. The speed of the football at different times is noted as given below (Football always remains in contact with hill) :
`{:("Time",,,"Instantaneous speed"),(0 sec,,," "0),(1 sec,,," "4m//s),(2 sec,,," "8 m//s),(3 sec,,," "16m//s),(4 sec,,," "24 m//s):}`
The correct shape of the hill is depicted in :

Options

A.
B.
C.
D.

Explanation

To determine the correct shape of the hill based on the speed of the football at different times, we will analyze the given data step by step.

### Step 1: Analyze the Speed Data
The speed of the football at different times is given as follows:
- At 0 seconds: 0 m/s
- At 1 second: 4 m/s
- At 2 seconds: 8 m/s
- At 3 seconds: 16 m/s
- At 4 seconds: 24 m/s

### Step 2: Calculate the Change in Speed
Next, we calculate the change in speed (Δv) for each second:
- From 0 to 1 second: Δv = 4 m/s - 0 m/s = 4 m/s
- From 1 to 2 seconds: Δv = 8 m/s - 4 m/s = 4 m/s
- From 2 to 3 seconds: Δv = 16 m/s - 8 m/s = 8 m/s
- From 3 to 4 seconds: Δv = 24 m/s - 16 m/s = 8 m/s

### Step 3: Analyze the Acceleration
Now, we can analyze the acceleration (a) which is the change in speed over time (Δv/Δt):
- From 0 to 1 second: a = 4 m/s / 1 s = 4 m/s²
- From 1 to 2 seconds: a = 4 m/s / 1 s = 4 m/s²
- From 2 to 3 seconds: a = 8 m/s / 1 s = 8 m/s²
- From 3 to 4 seconds: a = 8 m/s / 1 s = 8 m/s²

### Step 4: Determine the Nature of the Acceleration
From the calculations, we can see that the acceleration is not constant. It remains constant for the first two seconds and then increases in the next two seconds. This indicates that the hill's slope is changing.

### Step 5: Identify the Shape of the Hill
Given that the speed increases gradually at first and then accelerates more rapidly, the shape of the hill must allow for this increasing acceleration. The correct shape would be one that has a steeper slope after a certain point, allowing the football to gain speed more quickly.

### Conclusion
Based on the analysis, the correct shape of the hill is depicted in option 3, where the slope increases after a certain point, allowing for the increasing speed of the football.