A balloon is rising vertically upwards at constant speed of 10m/s. A stone is dropped when the balloon is at a height of 40m. Total distance covered by stone before reaching ground is-

Shouldnt the answer be 40m itself?

It's a common misconception to think that the distance a stone falls after being dropped from a balloon is simply the height from which it was released. However, in this scenario, we need to consider a few key factors to understand the total distance the stone travels before it hits the ground.

Understanding the Motion of the Stone

When the stone is dropped from the balloon, it is initially moving upward at the same speed as the balloon, which is 10 m/s. This means that the stone doesn't just fall straight down from the height of 40 meters; it first continues to move upward for a brief moment before starting its downward journey.

Breaking Down the Motion

To calculate the total distance the stone travels, we need to analyze two phases of its motion:

• Upward Motion: The stone will continue to rise for a short period until its upward velocity decreases to zero.
• Downward Motion: After reaching its peak height, the stone will then fall to the ground.

Calculating the Upward Motion

First, let's determine how long it takes for the stone to stop rising. The stone's initial velocity is 10 m/s, and it will decelerate due to gravity, which is approximately 9.81 m/s². We can use the following kinematic equation:

v = u + at

Where:

• v = final velocity (0 m/s at the peak)
• u = initial velocity (10 m/s)
• a = acceleration (-9.81 m/s², since it's acting downward)
• t = time taken to reach the peak

Rearranging the equation gives us:

0 = 10 - 9.81t

Solving for t:

t = 10 / 9.81 ≈ 1.02 seconds

Calculating the Height Gained

Next, we can find out how high the stone rises during this time using another kinematic equation:

s = ut + 0.5at²

Substituting the values:

s = (10)(1.02) + 0.5(-9.81)(1.02)²

Calculating this gives:

s ≈ 10.2 - 5.1 ≈ 5.1 meters

Total Distance Covered by the Stone

Now, we can find the total distance the stone travels:

• Distance traveled upward: 5.1 meters
• Distance fallen from the peak to the ground: 40 m + 5.1 m = 45.1 meters

Thus, the total distance covered by the stone before it reaches the ground is approximately 45.1 meters, not just the 40 meters from which it was dropped. This illustrates how the initial upward motion affects the total distance traveled.