A marble A is dropped vertically. Another identical marble B is projected horizontally from the same point at the same instant

Options

A. A will reachthe ground earlier than B.
B. B will reach the ground earlier than A.
C. Both A and B reach the ground at the same instant.
D. None of the above

Answer

To solve the problem, we need to analyze the motion of both marbles, A and B, when they are dropped or projected from the same height.

### Step 1: Understand the motion of marble A
Marble A is dropped vertically from a height \( H \). The time taken for marble A to reach the ground can be calculated using the formula for free fall:
\[ t_A = \sqrt{\frac{2H}{g}} \]
where \( g \) is the acceleration due to gravity.

### Step 2: Understand the motion of marble B
Marble B is projected horizontally from the same height \( H \). The horizontal motion does not affect the time it takes to fall vertically. The time taken for marble B to reach the ground is also given by the same formula:
\[ t_B = \sqrt{\frac{2H}{g}} \]

### Step 3: Compare the times
Since both marbles are falling from the same height \( H \) and are subject to the same gravitational acceleration \( g \), we can conclude:
\[ t_A = t_B \]

### Step 4: Conclusion
Both marbles A and B will reach the ground at the same instant. Therefore, the correct option is:
**Option C: both A and B reach the ground at the same instant.**

---