Calculating the Height of a Thrown Object Using Kinematics

Understanding the motion of an object subjected to gravitational forces is a fundamental part of physics. This article delves into the process of using kinematic equations to determine the height of an object thrown up at a specific speed after a given time. We will cover the example of an object thrown up with a speed of 50 m/s, and how to calculate its height after 7 seconds.

Kinematic Equations and Concepts

The motion of an object in the presence of constant acceleration, such as the acceleration due to gravity, can be analyzed using kinematic equations. These equations are derived from the basic laws of motion and are highly useful in solving problems related to the displacement, velocity, and acceleration of objects.

Problem Statement

Consider an object thrown up with an initial velocity of 50 m/s. We wish to determine its height from the ground after 7 seconds. This involves using the kinematic equation that relates the height, initial velocity, acceleration due to gravity, and time.

Applying the Kinematic Equation

The general formula for height as a function of time, initial velocity, and acceleration due to gravity is given by:

h v0 t 1/2 a t2

Where:

v0 is the initial velocity (50 m/s in this case) t is the time (7 s in this case) a is the acceleration due to gravity (-9.81 m/s2 since it acts downward)

Substituting the values into the equation:

h 50 m/s times; 7 s 1/2 times; -9.81 m/s2 times; 72 s2

Calculations

Let's break down the calculations step by step:

1. Calculate v0 t

50 m/s times; 7 s 350 m

2. Calculate 1/2 a t2

1/2 times; -9.81 m/s2 times; 72 s2

1/2 times; -9.81 times; 49

-240.645 m

Combining the Results

Now, combine the two results:

h 350 m - 240.645 m 109.355 m

Thus, the height of the object from the ground after 7 seconds is approximately 109.36 meters.

Additional Examples

Let's look at a second example where an object is shot upwards with an initial velocity of 135 m/s. We can use a similar technique to find its height after 7 seconds:

Given Data:

Initial velocity (v0) 135 m/s Time (t) 7 s Acceleration due to gravity (a) -9.81 m/s2

Kinematic Equation:

H v0 t - 1/2 a t2

135 times; 7 - 1/2 times; 9.81 times; 72

945 - 1/2 times; 9.81 times; 49

945 - 240.35 704.65 m

Conclusion

Understanding the motion of an object under the influence of gravity can be crucial in many real-life scenarios, from sports to engineering. By applying the principles of kinematic equations, we can accurately predict the height of an object at any given time during its trajectory.

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