A ball falls from the top of a high cliff. A second ball is thrown downward from the same height 4 secs later with an initial speed of 40m/s. If both balls hits the ground simultaneously. How high is the cliff?
0 = Hi - Vi t - 4.9 t^2
0 = Hi - 0 (t+4) - 4.9 (t+4)^2
0 = Hi - 40 t - 4.9 t^2
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Hi = 4.9 t^2 + 40 t
Hi = 4.9(t^2+8t+16)
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4.9 t^2+40 t = 4.9 t^2+39.2t +78.4
.8t = 78.4
t = 98
Hi = 40t+4.9t^2 =40*98+4.9(98^2)
=50,980 meters (30 miles? check arithmetic :)
To find the height of the cliff, we need to determine the time it takes for the first ball to hit the ground after being dropped and compare it with the time it takes for the second ball to hit the ground after being thrown downward.
Let's consider the motion of each ball separately:
1. For the ball that was dropped from the cliff:
When an object is dropped from rest, it accelerates due to gravity (9.8 m/s^2) in the downward direction. To calculate the time it takes for the ball to hit the ground, we can use the formula for the time of flight in free fall:
h = (1/2)gt^2
Here, h is the height of the cliff, g is the acceleration due to gravity, and t is the time taken for the ball to reach the ground. Since it was dropped from rest, its initial velocity is zero.
2. For the ball that was thrown downward:
The ball was thrown downward 4 seconds after the first ball was dropped. Its initial velocity is 40 m/s in the downward direction, and it also experiences acceleration due to gravity. We can use the following equation to determine the time it takes for the ball to reach the ground:
h = v0t + (1/2)gt^2
Here, h is the height of the cliff, v0 is the initial velocity of the ball, g is the acceleration due to gravity, and t is the time taken for the ball to reach the ground.
Since both balls hit the ground simultaneously, the time taken for both balls to reach the ground is the same. We can equate the two equations:
(1/2) * 9.8 * t1^2 = 40t2 + (1/2) * 9.8 * t2^2
Simplifying the equation, we can find the value of t1 in terms of t2.
Once we have the time t1, we substitute it back into the first equation to find the height h.
Finally, we can calculate the height of the cliff.