Calculating the Maximum Height and Total Time for a Stone Thrown Vertically

When a stone is thrown vertically upward, several factors determine how high it will travel before it starts to fall and how long it will take to return to the ground. This article explains the detailed steps and formulas used to calculate these parameters using basic mechanics and kinematic equations.

Introduction

The problem at hand involves throwing a stone vertically upward with an initial velocity of 49 meters per second (m/s). We need to determine two key factors: the maximum height the stone will reach before it starts to fall and the total time it will take for the stone to return to the ground.

Maximum Height Calculation

Using the kinematic equation for motion, we can determine the maximum height s reached by the stone. The relevant formula is:

v2 u2 - 2as

Variation and rearrangement of this formula can help us calculate the maximum height.

Step 1: Rearrange the Equation

Given:

v 0 m/s (at the maximum height) u 49 m/s (initial velocity) a -9.81 m/s2 (acceleration due to gravity, negative because it acts downward) s displacement (maximum height h)

Rearranging the formula to solve for s:

0 492 - 2×9.81s

2401 - 19.62s 0

s 2401 / 19.62 ≈ 122.9 meters

Time to Reach Maximum Height

Another important factor is the time taken by the stone to reach its maximum height. Using the kinematic equation for time:

v u at

Setting v 0 m/s:

0 49 - 9.81t

t 49 / 9.81 ≈ 4.98 seconds

Total Time to Reach the Ground

The total time for the stone to go up and come back down is twice the time to reach the maximum height. Therefore:

Total time ≈ 2 × 4.98 seconds ≈ 9.96 seconds

Summary

Based on the calculations:

Maximum height (reached by the stone) ≈ 122.9 meters Total time to reach the ground ≈ 9.96 seconds

Additional Insights

Another way to approach this problem is by using alternative formulas derived from the conservation of energy and kinematic principles.

Height Formula:

A common formula to use for the maximum height H is:

H u2 / (2g)

Given:

u 49 m/s g 9.81 m/s2

H 492 / (2 × 9.81) 2401 / 19.62 ≈ 122.5 meters

This confirms our earlier calculation with a slight variation due to rounding.

Time to Maximum Height:

Using the time formula:

T1 u / g

T1 49 / 9.81 ≈ 4.98 seconds

Total time ≈ 2 × 4.98 ≈ 9.96 seconds

Conclusion and Final Thoughts

To summarize, the stone is thrown vertically upward with an initial velocity of 49 m/s. By using the appropriate kinematic equations and considering the principles of conservation of energy, we can determine the maximum height it will reach and the total time it will take for the stone to return to its starting point.

These calculations are crucial for understanding the dynamics and mechanics of projectile motion, which are widely used in various fields such as physics, engineering, and everyday problem-solving scenarios.