A student performs an experiment to determine how the range of a ball depends on the velocity with which it is projected. The "range" is the distance between the points where the ball lends and from where it was projected, assuming it lands at the same height from which it was projected. It each trial, the student uses the same baseball, and launches it at the same angle. Table shows the experimental results.
`|{:("Trail","Launch speed" (m//s),"Range"(m)),(1,10,8),(2,20,31.8),(3,30,70.7),(4,40,122.5):}|`
Based on this data, the student then hypothesizes that the range, R, depends on the initial speed `v_(0)` according to the following equation : `R=Cv_(0)^(n)`, where C is a constant and n is another constant.
Based on this data, the best guess for the value of n is :-

Options

A. `1/2`
B. `1`
C. `2`
D. `3`

Explanation

To determine the best guess for the value of \( n \) in the equation \( R = C v_0^n \) based on the experimental data, we will follow these steps:

### Step 1: Write down the experimental data
The experimental data provided is as follows:

| Trial | Launch Speed \( v_0 \) (m/s) | Range \( R \) (m) |
|-------|-------------------------------|-------------------|
| 1 | 10 | 8 |
| 2 | 20 | 31.8 |
| 3 | 30 | 70.7 |
| 4 | 40 | 122.5 |

### Step 2: Set up the equation for each trial
According to the hypothesis, the relationship between range \( R \) and launch speed \( v_0 \) can be expressed as:
\[ R = C v_0^n \]
We can set up equations for each trial based on this relationship.

For Trial 1:
\[ 8 = C \cdot 10^n \] (Equation 1)

For Trial 2:
\[ 31.8 = C \cdot 20^n \] (Equation 2)

### Step 3: Divide the equations to eliminate \( C \)
To eliminate \( C \), we can divide Equation 2 by Equation 1:
\[ \frac{31.8}{8} = \frac{C \cdot 20^n}{C \cdot 10^n} \]
This simplifies to:
\[ \frac{31.8}{8} = \frac{20^n}{10^n} \]
\[ \frac{31.8}{8} = 2^n \]

### Step 4: Calculate the left side
Now, we compute \( \frac{31.8}{8} \):
\[ \frac{31.8}{8} = 3.975 \]

### Step 5: Set up the equation for \( n \)
Now we have:
\[ 2^n = 3.975 \]

### Step 6: Solve for \( n \)
To find \( n \), we can express \( 3.975 \) in terms of powers of 2:
\[ 2^n \approx 4 \quad \text{(since \( 4 \) is close to \( 3.975 \))} \]
This implies:
\[ 2^n = 2^2 \]
Thus, we find:
\[ n = 2 \]

### Conclusion
The best guess for the value of \( n \) is \( 2 \).

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