Calculating the Time for a Stone Thrown Vertically Upward to Reach the Ground

In this article, we will determine the time it takes for a stone thrown vertically upward with an initial velocity of 20 m/s to reach the ground using the laws of physics and equations of motion under constant acceleration due to gravity.

Understanding the Concept of Vertical Upward Motion

When a stone is thrown vertically upward, it moves under the influence of gravity. We can use the equations of motion to determine the time it takes for the stone to reach its maximum height and then fall back to the ground. The key equations we will use are:

Final velocity (v) Initial velocity (u) Acceleration (a) × Time (t)Displacement (s) Initial velocity (u) × Time (t) frac{1}{2} Acceleration (a) × Time^2 (t^2)Final velocity (v)^2 Initial velocity (u)^2 2 × Acceleration (a) × Displacement (s)

Step-by-Step Calculation

Given:- Initial velocity (u 20 , text{m/s}) upward.- Acceleration due to gravity (a -9.81 , text{m/s}^2) (negative because it acts downward).- Final velocity at the peak (v 0 , text{m/s}).Step 1: Time to reach the peak

At the peak, the stone's velocity is 0. We use the equation:

[v u at]

Setting [v 0] to find the time:

[0 20 - 9.81t]

Solving for [t]:

[9.81t 20]

[t approx frac{20}{9.81} approx 2.04 , text{seconds}]

Since the motion is symmetric, the time to fall from the peak back to the ground is also 2.04 seconds.

Total Time of Flight

The total time of flight is twice the time to reach the peak:

[text{Total time} 2t 2 times 2.04 approx 4.08 , text{seconds}]

Additional Examples and Solutions

Let's consider another example:

Vi 25 m/s, g 10 m/s^2

The time to reach the maximum height can be calculated as follows:

[0 25 - 10t]

Therefore:

[10t 25]

[t frac{25}{10} 2.5 , text{seconds}]

The time to rise to the maximum height is the same as the time to fall back down, so the total time of flight is:

[text{Total time} 2t 2 times 2.5 5 , text{seconds}]

Another way to solve this problem is by considering the symmetric nature of the motion:

In its upward journey, the stone moves against gravity (speed decreases until it becomes 0), then it starts moving downwards. The time needed for the initial upward speed to become 0 is given by:

[t frac{text{initial speed}}{g} frac{20}{9.81} approx 2.04 , text{seconds}]

After stopping momentarily, the stone travels downwards and will reach the ground in 2.04 seconds. Hence, the total time taken:

[text{Total time} 2.04 times 2.04 approx 4.08 , text{seconds}]

Conclusion: The stone will take approximately 4.08 seconds to reach the ground after being thrown upwards with an initial velocity of 20 m/s.

Summary of Key Points

The time to reach the maximum height is the same as the time to fall back to the total time of flight is twice the upward flight acceleration due to gravity is constant (-9.81 m/s2).The equations of motion can effectively solve various similar problems within the realm of vertical projectile motion.

Understanding and applying these principles can help in solving a wide range of physics problems related to projectile motion.