When a ball is thrown up vertically with velocity `v_0`, it reaches a maximum height of h. If one wishes to triple the maximum height then the ball should be thrown with velocity

Options

A. `sqrt3 v_0`
B. `3 v_0`
C. `9 v_0`
D. `3/2 v_0`

Explanation

To solve the problem of determining the velocity required to triple the maximum height reached by a ball thrown vertically upwards, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between initial velocity and maximum height**:
The maximum height \( h \) reached by a ball thrown upwards with an initial velocity \( v_0 \) is given by the formula:
\[ h = \frac{v_0^2}{2g} \]
where \( g \) is the acceleration due to gravity.

2. **Express the new height**:
If we want to triple the maximum height, the new height \( h' \) will be:
\[ h' = 3h \]

3. **Substitute the expression for height**:
Using the formula for maximum height, we can express \( h' \) in terms of the new initial velocity \( v \):
\[ h' = \frac{v^2}{2g} \]
Therefore, we have:
\[ 3h = \frac{v^2}{2g} \]

4. **Substitute the original height into the new height equation**:
We can substitute \( h = \frac{v_0^2}{2g} \) into the equation for \( h' \):
\[ 3 \left(\frac{v_0^2}{2g}\right) = \frac{v^2}{2g} \]

5. **Simplify the equation**:
By multiplying both sides by \( 2g \), we get:
\[ 3v_0^2 = v^2 \]

6. **Solve for the new initial velocity \( v \)**:
Taking the square root of both sides gives:
\[ v = \sqrt{3} v_0 \]

### Final Answer:
Thus, to triple the maximum height, the ball should be thrown with a velocity of:
\[ v = \sqrt{3} v_0 \]