Question
A stone A is thrown vertically upward at 10m/sec. Another stone b is thrown vertically upward at 12m/sec after 1 second at an initial point of B is 2 m higher than the initial point of A.
At what height that two stones have the same elevation?
At what height that two stones have the same elevation?
Answers
We can start by using the kinematic equation for the height of a vertically thrown object:
h = vi*t - 0.5*g*t^2
where h is the height, vi is the initial velocity, t is the time, and g is the acceleration due to gravity (9.8 m/s^2).
For stone A, we have:
hA = 10t - 0.5*9.8*t^2
For stone B, we have:
hB = 12(t-1) - 0.5*9.8*(t-1)^2 + 2
Note that we subtract 1 second from the time for stone B since it is thrown after 1 second, and we add 2 meters to account for the initial height difference between A and B.
To find the height at which the two stones have the same elevation, we need to find the value of t that makes hA = hB:
10t - 0.5*9.8*t^2 = 12(t-1) - 0.5*9.8*(t-1)^2 + 2
Simplifying and rearranging, we get:
4.9t^2 - 14t + 8 = 0
Using the quadratic formula, we get:
t = (14 ± sqrt(196 - 4*4.9*8)) / (2*4.9)
t ≈ 2.7 seconds or t ≈ 0.6 seconds
Note that we discard the negative solution since time cannot be negative. Therefore, the two stones have the same elevation at around t = 2.7 seconds.
To find the height at this point, we can plug t = 2.7 seconds into either hA or hB:
hA = 10*2.7 - 0.5*9.8*(2.7)^2 ≈ 37.4 meters
hB = 12(2.7-1) - 0.5*9.8*(2.7-1)^2 + 2 ≈ 37.4 meters
Therefore, the two stones have the same elevation at a height of approximately 37.4 meters.
h = vi*t - 0.5*g*t^2
where h is the height, vi is the initial velocity, t is the time, and g is the acceleration due to gravity (9.8 m/s^2).
For stone A, we have:
hA = 10t - 0.5*9.8*t^2
For stone B, we have:
hB = 12(t-1) - 0.5*9.8*(t-1)^2 + 2
Note that we subtract 1 second from the time for stone B since it is thrown after 1 second, and we add 2 meters to account for the initial height difference between A and B.
To find the height at which the two stones have the same elevation, we need to find the value of t that makes hA = hB:
10t - 0.5*9.8*t^2 = 12(t-1) - 0.5*9.8*(t-1)^2 + 2
Simplifying and rearranging, we get:
4.9t^2 - 14t + 8 = 0
Using the quadratic formula, we get:
t = (14 ± sqrt(196 - 4*4.9*8)) / (2*4.9)
t ≈ 2.7 seconds or t ≈ 0.6 seconds
Note that we discard the negative solution since time cannot be negative. Therefore, the two stones have the same elevation at around t = 2.7 seconds.
To find the height at this point, we can plug t = 2.7 seconds into either hA or hB:
hA = 10*2.7 - 0.5*9.8*(2.7)^2 ≈ 37.4 meters
hB = 12(2.7-1) - 0.5*9.8*(2.7-1)^2 + 2 ≈ 37.4 meters
Therefore, the two stones have the same elevation at a height of approximately 37.4 meters.
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