Question

A ball is projected horizontally from the top of a tower with a velocity of \( 10 \text{ m/s} \). If it hits the ground \( 2 \text{ seconds} \) later, what is the height of the tower? (Take \( g = 10 \text{ m/s}^2 \))

• A. \( 20 \text{ m} \)
• B. \( 10 \text{ m} \)
• C. \( 40 \text{ m} \)
• D. \( 25 \text{ m} \)

Hint:

In horizontal projection problems: \[ \text{Horizontal motion} \rightarrow \text{constant velocity} \] \[ \text{Vertical motion} \rightarrow \text{free fall under gravity} \] The height depends only on vertical motion and time of flight.

Answer 1

The Correct Option is A

Step 1 : Understanding the Question:
The question presents a classic scenario of horizontal projectile motion from a height. A ball is projected horizontally, meaning it has zero initial velocity in the vertical direction. Because horizontal and vertical motions are independent, the vertical descent is a simple case of free fall under gravity. Thus, we only need to look at the vertical components of displacement and acceleration.
Step 2 : Key Formulas and Approach:
We use the second equation of motion for vertical displacement:
\[ h = u_y t + \frac{1}{2} g t^2 \] Since the initial velocity is purely horizontal, the initial vertical velocity component \( u_y \) is zero:
\[ u_y = 0 \] By substituting \( u_y = 0 \), the height equation simplifies directly to:
\[ h = \frac{1}{2} g t^2 \] Our approach is to solve this vertical kinematics equation using the given flight time and the acceleration of gravity.
Step 3 : Detailed Solution:

List the given values: initial horizontal velocity \( u_x = 10 \text{ m/s} \), flight time \( t = 2 \text{ s} \), and acceleration due to gravity \( g = 10 \text{ m/s}^2 \).

Note that the horizontal velocity does not affect the time of descent or vertical displacement.

Write down the vertical displacement equation with the initial vertical velocity \( u_y = 0 \):
\[ h = 0 \cdot t + \frac{1}{2} g t^2 \]
Substitute \( g = 10 \text{ m/s}^2 \) and \( t = 2 \text{ s} \) into the simplified height formula:
\[ h = \frac{1}{2} \times 10 \times (2)^2 \]
Calculate the final numeric value to find the height of the tower:
\[ h = 5 \times 4 = 20 \text{ m} \]
Step 4 : Final Answer:
The height of the tower is \( 20 \text{ m} \), which corresponds to option (A).
\[ \boxed{20\text{ m}} \]