Displacement Calculation for Vertically Thrown Objects - An In-Depth Analysis

In physics, understanding the motion of an object thrown vertically upward is a fundamental concept. This tutorial provides a detailed look at the calculations needed to determine the displacement of such an object over a specific period of time. Using the provided values, we will walk through the mathematical steps to find the displacement of an object after 4 seconds.

Understanding the Motion and Key Concepts

When an object is thrown vertically upward, it moves under the influence of gravity. The motion is governed by several key parameters:

Initial velocity (v0) Gravitational acceleration (g -9.81 m/s2) Time (t) Displacement (d)

The initial velocity here is 30 m/s, and we are interested in calculating the displacement after 4 seconds. The negative sign for the gravitational acceleration indicates that it acts in the opposite direction to the initial velocity.

Applying Kinematic Equations

The kinematic equation for displacement is given by:

d  v0 t - frac{1}{2} g t^2

Substituting the given values:

v0 30 m/s t 4 s g -9.81 m/s2

Let's calculate the displacement step-by-step:

Calculate the first term: v0 t 30 m/s * 4 s 120 m

Calculate the second term using the formula (frac{1}{2} g t^2): (frac{1}{2} * -9.81 m/s^2 * (4 s)^2 -78.48 m

Combine the terms: d 120 m - 78.48 m 41.52 m

Thus, the displacement after 4 seconds is approximately 41.52 meters upward.

Verification Through Other Calculations

To further validate the calculation, we can use the following equations:

Velocity at any time (v v0 - g t) Displacement (s v0 t - frac{1}{2} g t^2) Maximum height (ymax (frac{v^20}{2g})) Time of flight (tflight (frac{2 v0}{g}))

Using these equations, we can confirm the following:

Velocity after 4 seconds: v 30 m/s - 9.8 m/s^2 * 4 s 9.2 m/s (downward) Displacement after 4 seconds: s 30 m/s * 4 s - 4.9 m/s^2 * (4 s)^2 41.6 m (upward) Maximum height: ymax (frac{30^2}{2 * 9.81}) 45.82 m Time of flight: tflight (frac{2 * 30}{9.81}) 6.12 s

Conclusion

In summary, the displacement of an object thrown vertically upward with an initial velocity of 30 m/s after 4 seconds is approximately 41.52 meters upward. This result is consistent with the various methods and equations used to verify the motion. Understanding these calculations is crucial for anyone studying mechanics and physics.

Related Topics

Kinematic Equations for Vertical Motion Calculating Displacement and Velocity .projectiles motion in physics