A ball is thrown vertically upward with a velocity u from the top of a tower. If it strikes the ground with velocity 3u, the time taken by the ball to reach the ground is

Options

A. `(u)/(g)`
B. `(2u)/(g)`
C. `(3u)/(g)`
D. `(4u)/(g)`

Answer

To solve the problem step by step, we will use the equations of motion under uniform acceleration. Here’s how we can approach it:

### Step 1: Understand the problem
A ball is thrown vertically upward with an initial velocity \( u \) from the top of a tower and strikes the ground with a final velocity of \( 3u \). We need to find the time taken by the ball to reach the ground.

### Step 2: Set up the equation
We will use the first equation of motion:
\[ v = u + at \]
where:
- \( v \) = final velocity
- \( u \) = initial velocity
- \( a \) = acceleration
- \( t \) = time

### Step 3: Define the variables
- Initial velocity \( u \) (upward direction)
- Final velocity \( v = -3u \) (downward direction, hence negative)
- Acceleration \( a = -g \) (acceleration due to gravity, also downward)

### Step 4: Substitute the values into the equation
Substituting the values into the equation:
\[ -3u = u - gt \]

### Step 5: Rearrange the equation
Rearranging the equation gives:
\[ -3u - u = -gt \]
\[ -4u = -gt \]

### Step 6: Solve for time \( t \)
Dividing both sides by \( -g \):
\[ t = \frac{4u}{g} \]

### Conclusion
The time taken by the ball to reach the ground is:
\[ t = \frac{4u}{g} \]

### Final Answer
Thus, the answer is \( \frac{4u}{g} \), which corresponds to option d.

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