The Question is from neet-chemistry atomic-structure horizontal-projectile, and it was created by the author Mr. S M HUSSAIN

Question

A ball is thrown horizontally from a cliff edge with velocity $v$.  Another ball is dropped vertically from the same point at the same instant.  Neglecting air resistance, which statement is TRUE about the time taken for each ball to reach the ground below?

Mr. S M HUSSAIN · Accepted Answer

## CORE INFORMATION
✓ **Answer**: C
## CONCEPT FRAMEWORK
 **Primary Concept**: Projectile motion and free fall under gravity.
 **Related Topics**: Kinematics, gravity, independent motion of horizontal and vertical components.
 **Prerequisites**: Understanding of basic kinematic equations, particularly those related to free fall.
## SOLUTION PATHWAY
**Given Information**:
  Two balls are released from the same height at the same time. One is thrown horizontally, the other is dropped vertically.
  Air resistance is negligible.
**Method & Steps**:
 1. Analyze the vertical motion of both balls.
  Reasoning: The vertical motion of both balls is governed solely by gravity. The horizontal velocity of the thrown ball does not affect its vertical motion.
  Formula/Rule:  $h = v_{0y}t + \frac{1}{2}gt^2$, where $h$ is the height, $v_{0y}$ is the initial vertical velocity, $g$ is the acceleration due to gravity, and $t$ is the time.
 2. Apply the equation to both balls.
  Working: For the dropped ball, $v_{0y} = 0$. For the thrown ball, $v_{0y} = 0$ as well (initial vertical velocity is zero).  Therefore, the equation for both balls simplifies to $h = \frac{1}{2}gt^2$.  Solving for $t$, we get $t = \sqrt{\frac{2h}{g}}$. Since $h$ and $g$ are the same for both balls, the time taken for both to reach the ground is the same.
  Key Point: The horizontal velocity of the thrown ball is irrelevant to the time it takes to fall vertically.
 3. State the conclusion.
  Result: Both balls hit the ground at the same time.
  Verification: The solution is verified by the fact that the vertical motion is independent of the horizontal motion in projectile motion when air resistance is neglected.
 ## OPTION ANALYSIS
A :x: The horizontal velocity does not affect the vertical fall time.
B :x:  The vertical motion is independent of the horizontal motion.
C ✓: Both balls experience the same vertical acceleration due to gravity and start from the same height with the same initial vertical velocity (zero).
D :x:  Mass does not affect the time taken for an object to fall under gravity (neglecting air resistance).
 ## LEARNING AIDS
 **Common Mistakes**:
  1. Assuming the horizontal velocity affects the vertical fall time: This is a common misconception. The two motions are independent.
  2. Forgetting to consider the initial vertical velocity: If there was an initial upward or downward vertical velocity, it would have to be incorporated in the equation.
 **Quick Tips**:
  1. Remember that the vertical and horizontal components of projectile motion are independent when air resistance is negligible.
  2. Visualize the situation to understand that both balls fall under the same gravitational acceleration.

Mr. S M HUSSAIN · Answer

The thrown ball hits the ground first.

Mr. S M HUSSAIN · Answer

The dropped ball hits the ground first.

Mr. S M HUSSAIN · Answer

The correct Option is "Both balls hit the ground at the same time."

Mr. S M HUSSAIN · Answer

The time taken depends on the mass of the balls.

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