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Exploring the Physics of Two Balls Thrown Upward: Finding the Time of Intersection
Exploring the Physics of Two Balls Thrown Upward: Finding the Time of Intersection
In this article, we will explore the physics of two balls thrown upward, focusing on the time at which they will meet. Specifically, we will delve into the mathematical and physical concepts that make this problem solvable, using SUVAT equations and simple mechanics.
Problem Statement
The problem at hand involves two balls: one ball is thrown vertically upward with an initial velocity of 96 m/s, and another ball, identical in every aspect, is thrown 4 seconds later with the same initial velocity. Our goal is to determine the time at which these two balls will meet.
Understanding the Problem
The scenario described involves two objects with identical characteristics (velocity, mass, initial position, and gravitational acceleration) but launched at different times. To solve this problem, we will employ the SUVAT (Up, Vertical, Slope, And Time) equations, which are commonly used in kinematics to describe the motion of objects under constant acceleration.
Solution
1. **Initial Velocity and Acceleration:**
The gravitational acceleration ( g ) is given as (-9.8 , text{m/s}^2). The initial velocity ( v_0 ) for both balls is 96 m/s.
2. **Position and Velocity of the First Ball:**
Let's analyze the position and velocity of the first ball at the time the second ball is thrown. The time elapsed since the first ball was thrown is 4 seconds.
The position of the first ball after 4 seconds is given by:
$$ s_1 frac{1}{2}gt^2 v_0 t $$
Here, ( t 4 ) seconds, ( v_0 96 , text{m/s} ), and ( g -9.8 , text{m/s}^2 ).
$$ s_1 frac{1}{2}(-9.8)(4^2) 96(4) $$
$$ s_1 -78.4 384 $$
$$ s_1 305.6 , text{meters} $$
The velocity of the first ball after 4 seconds is calculated as:
$$ v_1 v_0 gt $$
$$ v_1 96 - 9.8 times 4 $$
$$ v_1 96 - 39.2 $$
$$ v_1 56.8 , text{m/s} text{ upward} $$
3. **Position and Velocity of the Second Ball:**
The second ball is launched 4 seconds later. At this moment, its initial velocity is 96 m/s as well.
4. **Relative Velocity and Distance:**
The relative velocity of the two balls is the difference in their velocities:
$$ v_{text{relative}} v_1 - v_2 $$
$$ v_{text{relative}} 56.8 - 96 -39.2 , text{m/s} $$
The separation between the two balls is the distance the first ball has traveled minus the distance the second ball has traveled:
$$ s s_1 - s_2 $$
Since the second ball is just starting, its initial position is the same as the moment the first ball is at 4 seconds:
$$ s 305.6 - 0 305.6 , text{meters} $$
5. **Time of Intersection:**
The time it takes for the balls to meet can be found by considering the relative velocity and the initial distance between them:
$$ t frac{s}{v_{text{relative}}} $$
$$ t frac{305.6}{39.2} $$
$$ t approx 7.8 , text{seconds} $$
Since the second ball is launched 4 seconds later, the total time from the launch of the first ball is:
$$ t_{text{total}} 4 8 12 , text{seconds} $$
Therefore, the two balls will meet 12 seconds after the first ball is launched.
Conclusion
The problem of determining the time at which two balls thrown upward will meet involves understanding the kinematics of motion under constant acceleration. By applying the SUVAT equations, we can accurately predict the meeting time. The solution involves calculating the initial positions and velocities of both balls, determining the relative motion, and finally, solving for the time of intersection.