Finding the time when two stones cross paths

In summary, two stones are thrown simultaneously, one straight upward from the base of a cliff and the other straight downward from the top of the cliff. The height of the cliff is 6.69 m and the stones are thrown with the same speed of 9.97 m/s. After what time do the stones cross paths? By setting a common origin at the base of the cliff, we can solve the equation D_A=D_B, which yields the displacement of rock A as -6.69 m [up] and rock B as 0 m. The two stones cross paths after 0.82 seconds.
  • #1
Ace.
52
0

Homework Statement


Two stones are thrown simultaneously, one straight upward from the base of a cliff and the other straight downward from the top of the cliff. The height of the cliff is 6.69 m. The stones are thrown with the same speed of 9.97 m/s. After what time do the stones cross paths?

Homework Equations


d = vt + 0.5at2

The Attempt at a Solution


distance traveled by both rock A and rock B combined is 6.69
DA + DB = 6.69

let the upwards direction be positive

DA: (top of cliff)

d = vit + 0.5at2
= (-9.97 m/s[up])t + 0.5(-9.8 m/s2[up])t2
= -9.97t - 4.9t2

DB: (base of cliff)
d = vit + 0.5at2
d = (9.97 m/s [up])t + 0.5(-9.8m/s2[up])t2
d = 9.97t - 4.9t2

DA + DB = 6.69
-9.97t - 4.9t2 + 9.97t - 4.9t2 = 6.69
-9.8t2 = 6.69 getting negative time...?
t = 0.82 s

the correct answer is 0.335 seconds. I think I am messing up my directions and signs for my vectors... any help?
 
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  • #2
Good start.
Three things…

1. Keep your numbers as variables. It's a lot easier to keep writing "g" than to keep writing "-9.8 m/s2[up]".
2. You said.

"distance traveled by both rock A and rock B combined is 6.69
DA + DB = 6.69"

THis is close, but not quite correct. We don't know that this is the case. It would work if A was going down the whole time and B was going up the whole time. We know A will be going down the whole time, but B will reach a maximum height and begin to fall.

Your equation should mathematically say " The height of the cliff is 6.69 m".
 
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  • #3
Missed my third point…

"-9.8t2 = 6.69 getting negative time…?"

Nope, you wouldn't get negative time. It's even weirder; You would have imaginary time!
 
  • #4
It's probably easier to think about if you pick a common origin for ##D_A## and ##D_B##. Pick the common origin ##D=0## at the base of the cliff. So ##D_A=-9.97t - 4.9t^2 + 6.69## and ##D_B=9.97t-4.9t^2##. Now just solve ##D_A=D_B##.
 
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  • #5
Dick said:
It's probably easier to think about if you pick a common origin for ##D_A## and ##D_B##. Pick the common origin ##D=0## at the base of the cliff. So ##D_A=-9.97t - 4.9t^2 + 6.69## and ##D_B=9.97t-4.9t^2##. Now just solve ##D_A=D_B##.

##D_A + 6.69 = D_B##

It took me a while, but I finally found meaning behind this equation...
displacement of rock A: ##D_A = -6.69 m [up]##, and B: ##D_B = 0 m.##

-6.69 + 6.69 = 0, brilliant!
 

FAQ: Finding the time when two stones cross paths

What is the concept of finding the time when two stones cross paths?

The concept of finding the time when two stones cross paths is based on the idea of determining the exact moment when two objects in motion intersect with each other. In this case, the objects are two stones that are moving in different directions and at different speeds, and the goal is to calculate the time when they will cross paths.

What factors affect the time when two stones cross paths?

The time when two stones cross paths is influenced by several factors, including the initial position and velocity of each stone, the distance between the two stones, and the forces acting upon them (such as gravity or air resistance). These factors can affect the trajectory and speed of the stones, ultimately determining the time when they will intersect.

How is the time when two stones cross paths calculated?

The time when two stones cross paths can be calculated using the principles of physics and mathematical equations, such as the equations for motion and projectile motion. By plugging in the values for the initial position, velocity, and acceleration of each stone, the time when they will cross paths can be determined.

What are some real-life applications of finding the time when two stones cross paths?

Finding the time when two stones cross paths is a useful concept in a variety of fields, including sports, engineering, and astronomy. For example, in sports like tennis or baseball, calculating the time when a ball will cross paths with a player or another object can help improve performance and strategy. In engineering, this concept can be used to predict collisions and avoid accidents. In astronomy, it can be used to track the movements of celestial bodies.

What are some challenges in finding the time when two stones cross paths?

One of the main challenges in finding the time when two stones cross paths is accurately measuring and accounting for all the variables that can affect their motion. This includes factors such as air resistance, friction, and external forces. Additionally, the calculations can become more complex when dealing with multiple objects or when the initial conditions are constantly changing. It also requires a strong understanding of physics and mathematical concepts to accurately calculate the time of intersection.

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