Determining the Time for a Stone to Return to Ground After Being Thrown Upward

Understanding the physics behind the motion of a stone thrown vertically upward is an essential topic in introductory physics. This article explains the process of calculating the time taken for a stone thrown with an initial velocity to return to the ground. Using various methods and formulas, we will explore the physics behind this problem and derive the time it takes for the stone to reach its maximum height and then return to the ground.

Initial Problem and Setup

A stone is thrown vertically upward with an initial velocity of 25 m/s. The objective is to determine the total time taken for the stone to return to the ground. We will approach this problem step-by-step using both kinematic equations and simpler methods.

Method 1: Using Kinematic Equations

Step 1: Determine the Time to Reach Maximum Height

The stone reaches the maximum height where its final velocity is 0 m/s. We can use the kinematic equation:

v u - gt

Where:

v 0 m/s (final velocity at highest point) u 25 m/s (initial velocity) g 9.81 m/s2 (acceleration due to gravity) t time to reach maximum height

Rearranging the equation for t:

0 25 - 9.81t

9.81t 25

t 25 / 9.81 ≈ 2.55 seconds

Step 2: Determine the Total Time for the Round Trip

The time taken to go up is 2.55 seconds, and due to symmetry, the time taken to come back down is also 2.55 seconds. Therefore, the total time T is:

T 2 × t 2 × 2.55 ≈ 5.10 seconds

Conclusion: The total time for the stone to return to the ground is approximately 5.10 seconds.

Method 2: Symmetrical Motion

Alternatively, we can solve this problem by recognizing that the motion is symmetrical. The time needed for the initial upward speed to become 0 is given by:

t initial speed / g 20 / 9.81 ≈ 2.04 seconds

After reaching the highest point, the stone travels back down in the same amount of time. Therefore, the total time is:

Total time 2.04 2.04 4.08 seconds

Alternative Formula: SUVAT for Acceleration Due to Gravity

Using the SUVAT (Suvat) equations, where s is the displacement, u is the initial velocity, v is the final velocity, a is the acceleration, and t is the time, we can derive a simpler formula:

y gt^2 - (gt)^2 / u

Where:

g is the standard gravity (approximately -9.80665 m/s^2) u is the initial vertical velocity (positive is 'up') y is the initial and final height (0 in this case)

By solving for t, we can find the time taken to return to the ground.

Key Takeaways

Understanding the time taken for a stone to return to ground involves using kinematic equations or recognizing the symmetrical nature of the motion. The standard gravity (g) is a crucial constant in these equations. Recognizing the symmetry of the motion can simplify the calculations.

By applying these methods, students and learners can deepen their understanding of Newton's laws and kinematics.