A research probe needs to be .5 meters under the ground. It is known that the probe must be traveling 140 meters per second at impact to reach a depth of .5 meters. If the probe is dropped from 10b 2 (b is constant) meters above the ground and the distance traveled is given by the function s(t) = 4.9t 2 . Find the value of b and the height the probe should be dropped from.

Question

A research probe needs to be .5 meters under the ground. It is known that the probe must be traveling 140 meters per second at impact to reach a depth of .5 meters. If the probe is dropped from 10b 2 (b is constant) meters  above the ground and the distance traveled is given by the function s(t) = 4.9t 2 . Find the value of b and the height  the probe should be dropped from.

Bytelearn · Accepted Answer

Understand the problem: Understand the problem. We need to find the value of b and the height from which the probe should be dropped so that it reaches an impact velocity of 140 meters per second to achieve a depth of 0.5 meters. The distance traveled by the probe as it falls is given by the function s(t) = 4.9t^2, where t is the time in seconds.Use kinematic equation: Use the kinematic equation for the final velocity. The final velocity (v) of the probe when it hits the ground can be found using the kinematic equation v^2 = u^2 + 2as, where u is the initial velocity (which is 0 since the probe is dropped), a is the acceleration due to gravity (9.8 m/s^2), and s is the distance traveled. We know that v = 140 m/s and a = 9.8 m/s^2, so we can solve for s.Calculate distance using final velocity: Calculate the distance s using the final velocity. Plugging the values into the equation: (140 m/s)^2 = 0 + 2 * (9.8 m/s^2) * s s = (140 m/s)^2 / (2 * 9.8 m/s^2) s = 19600 m^2/s^2 / 19.6 m/s^2 s = 1000 meters The distance s is the height from which the probe should be dropped.Relate distance to function: Relate the distance s to the given function s(t). We have the function s(t) = 4.9t^2, and we found that s = 1000 meters. We can set these equal to solve for t. 1000 meters = 4.9t^2 t^2 = 1000 meters / 4.9 m/s^2 t^2 = 204.08163 seconds^2 t = √204.08163 seconds^2 t ≈ 14.28 secondsFind value of b: Find the value of b from the given height expression. We are given that the probe is dropped from a height of 10b^2 meters. We know this height must be equal to the distance s, which is 1000 meters. 10b^2 = 1000 meters b^2 = 1000 meters / 10 b^2 = 100 meters b = √100 meters b = 10 meters

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