An object is thrown downward with an initial speed of 17 m/s from a height of 96 m above the ground. At the same instant, a second object is propelled vertically up from ground level with a speed of 22 m/s. At what height above the ground will the two objects pass each other? The acceleration of gravity is 9.8 m/s2 . Answer in units of m.
height1 = -4.9t^2 - 17t + 96
height2 = -4.9t^2 + 22t
when are they equal?
solve for t, sub back into either of the equations.
To find the height at which the two objects pass each other, we need to determine the time it takes for both objects to reach the same height.
First, let's calculate the time it takes for the object thrown downward to reach the ground. We can use the following kinematic equation:
h = ut + (1/2)gt^2
where:
h = height of the object (96 m)
u = initial velocity of the object (-17 m/s, negative because it's thrown downward)
g = acceleration due to gravity (-9.8 m/s^2)
t = time
Rearranging the equation, we get:
(1/2)gt^2 + ut - h = 0
Plugging in the values, we have:
(1/2)(-9.8)t^2 + (-17)t - 96 = 0
Now, let's calculate the time it takes for the object propelled upward to reach the same height. We can use the same equation, but this time the initial velocity of the object will be positive:
h = ut + (1/2)gt^2
where:
h = height of the object (unknown, let's call it x)
u = initial velocity of the object (22 m/s)
g = acceleration due to gravity (-9.8 m/s^2)
t = time
Rearranging the equation:
(1/2)gt^2 + ut - x = 0
Plugging in the values, we have:
(1/2)(-9.8)t^2 + (22)t - x = 0
Now, let's solve both equations simultaneously to find the time (t) and height (x) at which the two objects meet.
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Step 1: Solve the first equation for the time (t) using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 0.5*(-9.8), b = -17, and c = -96.
t = (-(-17) ± √((-17)^2 - 4*(0.5*(-9.8))*(-96))) / (2*(0.5*(-9.8)))
Simplifying the equation, we get:
t = (17 ± √(289 + 1960)) / (-9.8)
t ≈ 1.897 s (positive value)
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Step 2: Substitute the found value of t into the second equation to find the height (x) at the same time:
(1/2)(-9.8)*(1.897)^2 + (22)*(1.897) - x = 0
Simplifying the equation, we get:
-9.8*(1.897)^2 + 22*(1.897) - x = 0
Calculating the value of x, we have:
x ≈ 34.267 m
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Therefore, the two objects will pass each other at a height of approximately 34.267 m above the ground.