Question

You throw a ball of mass 2 kilograms into the air with an upward velocity of \(8 \mathrm{~m} / \mathrm{s}\). Find exactly the time the ball will remain in the air, assuming that gravity is given by \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\).

Short answer

The ball remains in the air for approximately 1.632 seconds.

Step-by-step solution

Understanding the Problem

We are tasked with finding out how long the ball remains in the air. This involves calculating the total time the ball takes to reach its peak height and return to the ground.

Identify Key Equations

The motion of the ball is governed by the kinematic equation for vertical motion: \[ v = u + at \]where \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is the acceleration due to gravity \(-9.8 \text{ m/s}^2\), and \(t\) is the time.

Calculate Time to Reach Peak Height

At the peak height, the final velocity \(v\) is 0. Substitute \(u = 8 \text{ m/s}\) and \(a = -9.8 \text{ m/s}^2\) into the equation to find \(t_{up}\) (time to reach peak height):\[ 0 = 8 - 9.8t_{up} \]Solving this gives:\[ t_{up} = \frac{8}{9.8} \approx 0.816 \text{ seconds} \]

Calculate Total Time in the Air

The total time the ball remains in the air is twice the time to reach the peak height, as the time to go up equals the time to come down. Therefore, the total time is:\[ T = 2 imes t_{up} = 2 imes 0.816 \approx 1.632 \text{ seconds} \]

Key concepts

Projectile Motion

Projectile motion involves objects that are launched into the air and influenced only by gravity. The motion can be broken into two components: horizontal and vertical. In our scenario, since the ball is thrown vertically, we deal primarily with vertical motion.
The object is initially propelled upward, rising to a peak where its velocity becomes zero before gravity pulls it back down. Key features of vertical projectile motion include:

• Initial velocity: This is the velocity at which the object is thrown or propelled. In our case, it's 8 m/s upward.
• Peak height: This is the highest point the object reaches, where vertical velocity halts momentarily.
• Total time in air: The sum of the time taken to ascend and descend.

Understanding these aspects helps in analyzing the motion and predicting outcomes like maximum height and total air time.

Acceleration due to Gravity

Gravity is a force pulling objects toward Earth's center, giving them an acceleration of approximately 9.8 m/s². This value is crucial in calculating the motion of airborne objects like our ball.
When a ball is thrown upward, gravity acts downward, slowing its upward motion until it stops for a split second at the peak. When it falls back down, gravity accelerates it downward.
Key points about acceleration due to gravity:

• It always acts downward, regardless of the object's motion direction.
• It's a constant force, meaning it doesn't change as the object moves.
• In our equations, it's often denoted as a negative value to indicate its direction.

This consistent acceleration allows us to use kinematic equations to calculate various aspects of motion, like time in the air and peak height.

Kinematic Equations

These are essential mathematical tools in analyzing motion under a constant acceleration, like gravitational acceleration. Kinematic equations relate the five key motion variables: initial velocity, final velocity, acceleration, time, and displacement.
In this exercise, the relevant kinematic equation is:\[ v = u + at \]Here's how it applies:

• Initial velocity (\(u\)): The velocity at the beginning of the observation period. Here, it's 8 m/s.
• Final velocity (\(v\)): The velocity at the end of the observation period. At the peak of the ball’s path, this is 0 m/s.
• Time (\(t\)): The duration over which the observation occurs. This can be calculated for the ascent, the descent, or the total time in the air.

Using these variables, kinematics helps us solve for unknowns like the time to reach the peak height and the total time in the air. By inserting known values into the equation, we determine these critical timings which help us understand the full scope of the projectile's motion.