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Question

Two balls are projected horizontally from the same height with different initial speeds. Neglecting air resistance, which statement is true about their times of flight?

VENKATESWARARAO · Accepted Answer

## CORE INFORMATION
✓ **Answer**: [Option C]
## CONCEPT FRAMEWORK
	**Primary Concept**: Projectile Motion
	**Related Topics**: Kinematics, Gravity
	**Prerequisites**: Understanding of vertical and horizontal motion in projectile motion
## SOLUTION PATHWAY
**Given Information**:
	* Two balls projected horizontally
	* Same initial height
	* Different initial horizontal speeds
	* Neglecting air resistance
**Method & Steps**:
	1.  Analyze the vertical motion:
		Reasoning: The time of flight for a projectile is determined solely by its initial vertical velocity and the acceleration due to gravity. Since both balls are projected horizontally, their initial vertical velocity is zero.
		Formula/Rule:  Vertical displacement:  $Δy = v_{0y}t + \frac{1}{2}gt^2$
	2. Calculate the time of flight:
		Working: Since the initial vertical velocity ($v_{0y}$) is zero for both balls and they fall from the same height ($Δy$), the time of flight (t) is dependent only on the acceleration due to gravity (g), which is constant for both. Therefore $Δy = \frac{1}{2}gt^2$ or $t = \sqrt{\frac{2Δy}{g}}$
		Key Point:  Horizontal velocity does not affect the time of flight. 
	3. Final Conclusion:
		Result:  Both balls will land at the same time.
		Verification: The time of flight is solely determined by the vertical motion and since both balls have the same initial vertical conditions, they take the same time to hit the ground. 
 ## OPTION ANALYSIS
[A] :x: The horizontal speed does not affect the time of flight; it only influences the horizontal distance traveled.
[B] :x: The horizontal speed does not affect the time of flight.
[C] ✓: This is correct as explained in the solution pathway.
[D] :x: The mass of the balls does not affect their time of flight when air resistance is neglected. 
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Assuming horizontal speed affects time of flight: This is a common mistake. Remember that horizontal and vertical motions are independent.
		2.[Error 2]: Forgetting to consider initial vertical velocity: Ensure you correctly determine initial vertical velocity, which is zero in this case. 
	**Quick Tips**:
		1.[Helpful shortcut]: The time of flight depends only on the vertical motion.
		2.[Memory aid]: Think of dropping a ball straight down and throwing a ball horizontally; both will hit the ground at the same time if released from the same height.

VENKATESWARARAO · Answer

The ball with the higher initial speed will land first.

VENKATESWARARAO · Answer

The ball with the lower initial speed will land first.

VENKATESWARARAO · Answer

The correct Option is "Both balls will land at the same time."

VENKATESWARARAO · Answer

The time of flight depends on the mass of the balls.

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