Kinematics of a Ball Thrown Vertically Upwards: Displacement and Velocity Calculations

In physics, the motion of a ball thrown vertically upwards can be analyzed using kinematic equations. This article focuses on calculating the displacement of a ball thrown vertically upwards with an initial speed of 23 meters per second (m/s) and the velocity of the ball 3 seconds after it was thrown. We'll use the kinematic equation for displacement under constant acceleration and the SUVAT (stroke, undershoot, velocity, acceleration, time) formulas to derive the necessary values.

Understanding the Problem

A ball is thrown vertically upwards with an initial speed of 23 meters per second. To find the displacement of the ball 3 seconds after it was thrown, we use the kinematic equation for displacement under constant acceleration:

s ut (1/2)at^2

Where:

s displacement in meters (m) u initial velocity in m/s a acceleration due to gravity, which acts downwards approximately -9.81 m/s2 t time in seconds (s)

Calculating Displacement

Substituting the given values into the equation:

s 23 m/s * 3 s (1/2) * (-9.81 m/s2) * (3 s)2

Step 1: Calculate the initial velocity term

Initial velocity term: 23 m/s * 3 s 69 m

Step 2: Calculate the acceleration term

Acceleration term: (1/2) * (-9.81 m/s2) * (3 s)2 -44.145 m

Step 3: Combine both terms to find the total displacement

s 69 m - 44.145 m 24.855 m

Therefore, the displacement of the ball 3 seconds after it was thrown is approximately 24.86 meters upwards.

Calculating Velocity After 3 Seconds

To find the velocity of the ball after 3 seconds, we use the SUVAT formula v u at, where:

u initial velocity (19.6 m/s, which was the velocity after 2 seconds of upward motion) a acceleration due to gravity (-9.81 m/s2) t time (3 s)

Substituting the values:

v 19.6 m/s (-9.81 m/s2) * 3 s

v 19.6 m/s - 29.4 m/s -9.8 m/s

The negative sign indicates that the velocity is downwards. The ball is moving 9.8 m/s downwards after 3 seconds.

Real-World Application

The calculations above can be applied to various real-world scenarios, such as measuring the distance covered by an object thrown vertically upwards. For example, if a ball is thrown upwards with an initial velocity of 30 m/s and we desire the distance covered after 4 seconds:

s ut - (1/2)gt^2

Using: u 30 m/s, t 4 s, g 9.8 m/s2

s 30 m/s * 4 s - (1/2) * 9.8 m/s2 * (4 s)2

s 120 m - 78.4 m 41.6 m

Hence, the ball covers a distance of 41.6 meters after 4 seconds.

Conclusion

Understanding and applying the kinematic equations is crucial for analyzing the motion of objects under constant acceleration. Whether calculating displacement, velocity, or distance covered, the SUVAT formulas and kinematic equations provide a robust framework for solving such problems. By mastering these concepts, students can better understand and apply the principles of physics in various practical scenarios.