Question

A ball is tossed vertically upward with an initial speed of \(26.4 \mathrm{~m} / \mathrm{s}\). How long does it take before the ball is back on the ground?

Short answer

Question: A ball is thrown vertically upward with an initial speed of 26.4 m/s. Calculate the total time taken by the ball to be back on the ground. Answer: The ball takes 5.38 seconds before it is back on the ground.

Step-by-step solution

Identify the given parameters and unknowns

We are given: Initial velocity (u) = \(26.4 \mathrm{~m/s}\) Final velocity at maximum height (v) = \(0 \mathrm{~m/s}\) //Since the ball will stop momentarily at the highest point Acceleration due to gravity (a) = \(-9.81 \mathrm{~m/s^2}\) // The gravitational force acts in the downward direction, opposite to the direction of the initial velocity, hence it is negative We need to find the time (t) it takes for the ball to be back on the ground.

Calculate the time taken to reach maximum height

Using the first equation of motion: $$v = u + at$$ We can solve for the time taken to reach maximum height (t1), when the ball stops momentarily (v = 0). $$0 = 26.4 + (-9.81) t1$$ Now, solve for t1: $$t1 = \frac{26.4}{9.81} = 2.69 \mathrm{~s}$$

Calculate the time taken to fall back to the ground

Since it's a symmetrical situation, the time taken for the ball to fall back to the ground will be the same as the time taken to reach its maximum height. So, the time taken to fall back to the ground (t2) will be the same as t1. Thus, $$t2 = 2.69 \mathrm{~s}$$

Calculate the total time before the ball is back on the ground

Now we can calculate the total time (t) by adding the time taken to reach maximum height (t1) and the time taken to fall back to the ground (t2): $$t = t1 + t2$$ $$t = 2.69 + 2.69 = 5.38 \mathrm{~s}$$ So, it takes the ball \(5.38 \mathrm{~s}\) before it is back on the ground.

Key concepts

Kinematics

Kinematics is a branch of mechanics focusing on the motion of objects without considering the forces causing the motion. It's a fundamental concept in physics, helping us describe how an object moves in terms of velocity, displacement, and time. In the context of projectile motion, kinematics helps us determine how objects like balls move through the air when thrown upward.

By understanding parameters such as initial velocity and acceleration, we can predict the behavior of an object over time.

• Key Elements of Kinematics: It mainly involves quantitative analysis using equations and qualitative analysis through graph interpretation.
• Displacement and Velocity: These quantities describe the change in position of an object and how fast it's moving. In the given exercise, the ball's initial velocity is a crucial factor.
• Acceleration: Constant acceleration due to gravity is a major player in free-fall motion, affecting how the object's speed changes over time.

Understanding kinematics allows us to predict outcomes like the time a ball takes to return to the ground in a vertical throw.

Equations of Motion

The Equations of Motion are key tools in physics for analyzing the movement of objects. These are used extensively in kinematics to relate variables such as velocity, acceleration, time, and displacement.

For a ball thrown upward as in this case, these equations help calculate how long the ball stays in the air.

• First Equation of Motion: The formula \( v = u + at \) is used to determine the final velocity, where \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time.
• Application: In the problem, this formula finds the time to reach maximum height by setting the final velocity at the peak to zero. This calculation gives the duration of ascent.
• Symmetry in Motion: It's important to note that for vertical projectile motions under gravity, the time to ascend is equal to the time to descend.

The congruence of ascent and descent times simplifies calculations, making the equations of motion invaluable for solving problems involving free-falling objects.

Free Fall

Free fall is a specific type of motion where gravity is the only force acting on a falling object. When we neglect air resistance, every object irrespective of its mass falls with the same acceleration, defined as \(g = 9.81 \mathrm{~m/s^2}\).

This concept is essential in the analysis of projectile motion, as seen in the exercise where the ball is tossed upward.

• Characteristics of Free Fall: In free fall, initial upward velocity reduces due to gravity until the object stops momentarily, then descends with increasing speed.
• Symmetrical Path: Objects in free fall have symmetrical motion, meaning the time to climb to the peak is equal to the time to descend back to the ground.
• Constant Acceleration: Acceleration remains constant at \(-9.81 \mathrm{~m/s^2}\) during the entire motion, affecting both ascent and descent.

Understanding free fall allows students to calculate how long an object takes to return to its starting point or the ground. This makes it a key concept when solving real-world physics problems.