Final answer:
To calculate the total time the stone takes to reach the ground when dropped from a vertically ascending helicopter, we use the kinematic equations to find the time to stop its ascent due to gravity and then the time it would take to free fall from its highest point to the ground.
Step-by-step explanation:
To determine how long the stone takes to reach the ground after being dropped from a helicopter ascending vertically at a speed of 19.6 m/s at a height of 156.8 m, we need to incorporate both the upwards motion imparted by the helicopter and the free fall motion of the stone. First, the stone will continue to move upward until its velocity reaches zero due to gravity's deceleration. Then it will fall down to the ground.
We will use the kinematic equations to solve this problem. The first part is calculating the time the stone takes to stop its ascent. The initial velocity is +19.6 m/s (upwards), the acceleration due to gravity is -9.8 m/s² (downwards), and we set the final velocity to 0 m/s to find the time taken to come to a stop.
The equation for this is v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time. After finding the time it takes to stop, we then calculate the distance it has traveled during this time using the same kinematic equation s = ut + 0.5at², where s is the distance. The stone will have ascended further by this distance.
Once the stone has stopped its ascent, it will begin to fall from its highest point. Suppose its highest point above ground is H, then the total distance it will fall is H + 156.8 m. We now treat this as a separate motion of free fall from rest and use s = ut + 0.5at² again with u = 0, to find the time it takes to hit the ground.
Combining the time taken to stop ascending and the time taken to fall to the ground gives us the total time the stone is in the air. This method neglects any air resistance, which is a reasonable approximation for such calculations.