Final answer:
The maximum height above the robot that the stone reached is 7.056 meters, calculated using the kinematic equation for upward projectile motion under gravity, and considering the stone's total travel time to be 2.4 seconds.
Step-by-step explanation:
The problem concerning the robot that catches a stone after firing it straight upward can be solved using the kinematic equations of motion under the influence of gravity. When the stone is thrown upwards, it will decelerate under gravity until it reaches its peak height where its velocity is zero, and then it will accelerate downwards until caught by the robot.
Assuming upward motion is positive, the initial velocity (u) when the stone is thrown is unknown. The acceleration due to gravity (a) is -9.8 m/s2 (negative because the acceleration due to gravity is downwards). The time it takes for the stone to return to the robot's hand (t) is given as 2.4 seconds.
The total time for the journey can be divided into two equal parts: the time taken to reach the maximum height and the time taken to descend from the maximum height. Since these times are equal, the time to reach the maximum height is half of the total time: t/2 = 2.4 s / 2 = 1.2 s.
Using the kinematic equation h = ut + (1/2)at2, where h is the height, we can determine the maximum height reached by the stone. At maximum height, u = 0, therefore we can simplify the equation to h = (1/2)at2.
Plugging the values we have:
h = (1/2)(-9.8 m/s2)(1.2 s)2
h = (1/2)(-9.8 m/s2)(1.44 s2)
h = -7.056 m
Since height cannot be negative, the absolute value is taken, and the maximum height (h) above the robot that the stone reached is 7.056 meters.