A stone thrown vertically upwards with a speed of 12m/s from the edge of Cliff 70m high. Calculate the time the stone takes to hit the bottom of the cliff.
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To calculate the time it takes for the stone to hit the bottom of the cliff, we can use the equation of motion.
1. First, we need to determine the initial velocity and final velocity of the stone when it hits the bottom of the cliff. Since the stone is thrown vertically upwards, the initial velocity is given as +12 m/s (positive sign indicates upward direction). The final velocity when the stone hits the bottom of the cliff is 0 m/s since it comes to rest.
2. Next, we need to know the acceleration acting on the stone. In this case, we have to consider the acceleration due to gravity, which is always -9.8 m/s^2 (negative sign indicates downward direction).
3. By using the second equation of motion:
v^2 = u^2 + 2as,
where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement.
Since the final velocity, v, is 0 m/s, and the initial velocity, u, is +12 m/s, and the acceleration, a, is -9.8 m/s^2, and the displacement, s, is -70 m (negative sign indicates downward displacement), we can rewrite the equation as:
0^2 = 12^2 + 2(-9.8)s
Simplifying further, we get:
0 = 144 - 19.6s
Rearranging the equation, we have:
19.6s = 144
Dividing both sides of the equation by 19.6, we find:
s = 144/19.6
Solving this equation, we get:
s ≈ 7.35 m
4. Now that we know the displacement (s) and initial velocity (u), we can use the third equation of motion:
s = ut + (1/2)at^2
Substituting the known values:
-70 = (12)t + (1/2)(-9.8)t^2
Simplifying further, we have:
-70 = 12t - 4.9t^2
Rearranging the equation, we get:
4.9t^2 - 12t - 70 = 0
We can solve this quadratic equation using the quadratic formula. The formula is:
t = (-b ± sqrt(b^2 - 4ac))/(2a)
For this equation, a = 4.9, b = -12, and c = -70. Substituting these values into the quadratic formula, we have:
t = (-(-12) ± sqrt((-12)^2 - 4(4.9)(-70))) / (2(4.9))
Simplifying further, we get:
t = (12 ± sqrt(144 + 1372)) / 9.8
t = (12 ± sqrt(1516)) / 9.8
Evaluating the square root, we have:
t ≈ (12 ± 38.96) / 9.8
Hence, we have two possible solutions:
t1 ≈ (12 + 38.96) / 9.8 ≈ 5.15 seconds (ignoring the negative value since time cannot be negative in this context)
t2 ≈ (12 - 38.96) / 9.8 ≈ -2.78 seconds (discarding the negative value)
Therefore, the time it takes for the stone to hit the bottom of the cliff is approximately 5.15 seconds.