Problem 89 - Fundamentals of Physics

Question

A certain juggler usually tosses halls vertically to a height \(H\). To what height must they be tossed if they are to spend twice as much time in the air?

Step-by-Step Solution

Answer

Double the original height, \( H' = 2H \).

Step 1: Understand the Time in Air Formula

The time a ball spends in the air when thrown vertically is given by the formula \( t = \frac{2v_0}{g} \), where \( v_0 \) is the initial velocity and \( g \) is the acceleration due to gravity. The total time in the air is proportional to the initial velocity. Let's denote the original time the ball is in the air as \( t_1 \).

Step 2: Determine New Time Requirement

We are asked to determine the height such that the time in the air, \( t_2 \), is twice the original time \( t_1 \). Therefore, \( t_2 = 2t_1 \).

Step 3: Relate Initial Velocity to Toss Height

The maximum height \( H \) that a ball reaches can be written as \( H = \frac{v_0^2}{2g} \). This equation stems from the kinematic equation for vertical motion where final velocity at the peak is zero.

Step 4: Calculate New Initial Velocity

If \( t_2 = 2t_1 \), then the new initial velocity \( v_0' \) must be \( 2v_0 \), because \( t_2 = \frac{2v_0'}{g} = 2 \times \frac{2v_0}{g} \).

Step 5: Find the New Height

Substitute \( v_0' = 2v_0 \) into the height equation: \( H' = \frac{(2v_0)^2}{2g} = \frac{4v_0^2}{2g} = 2 \frac{v_0^2}{g} \). Therefore, the balls must be tossed to a height double the original, \( H' = 2H \).

Key Concepts

Vertical Motion

When we talk about vertical motion, we are focusing on the movement of objects thrown upwards or dropped straight down under the influence of gravity. Imagine tossing a ball straight into the air. At first, it rises until it reaches its peak height, where its speed momentarily becomes zero. Then, gravity takes over and pulls it back down.

This motion can be broken down into two main phases: the ascent and the descent. During the ascent, the ball slows down until it stops at the maximum height. On the descent, it speeds up as it falls back to the starting point. The entire time the ball is in the air is often referred to as the time of flight.

• Ascent: Decreasing speed until zero at the peak.
• Descent: Increasing speed back down.

Understanding this symmetry helps to comprehend how long an object remains in the air and how high it can go depending on its initial speed.

Kinematic Equations

Kinematic equations are essential tools in physics. They allow us to predict future motion, given initial conditions. There are four main equations, but the one we are interested in for vertical motion is:\[ H = \frac{v_0^2}{2g} \]This formula helps us determine the maximum height \( H \) an object can reach when thrown upward with an initial velocity \( v_0 \). It assumes constant acceleration due to gravity \( g \), which simplifies our calculations.

Another crucial equation is related to the total time an object spends in the air, given by:\[ t = \frac{2v_0}{g} \]

With these equations, we can make predictions, like in the juggler problem. If you want the ball to stay in the air twice as long, you need to double the height it reaches. This means recalculating the initial velocity and understanding how each factor contributes to the overall motion.

Acceleration due to Gravity

Gravity is a constant force that affects all objects near the Earth's surface. It acts to pull objects downward and determines how fast they fall. The acceleration due to gravity is denoted as \( g \) and is approximately \( 9.81 \, m/s^2 \) on Earth.

This acceleration is what causes the symmetry in vertical motion. When an object is tossed upward, gravity slows it down until it stops at its peak. On the way down, gravity accelerates the object back to the ground at the same rate. Thus, both the ascent and descent are influenced by this constant acceleration.

• It governs how quickly velocity changes over time.
• Plays a crucial role in calculating the time of flight and maximum height.

For all calculations involving motion, such as those in kinematic equations, gravity remains a vital factor. Its constant value allows us to predict and understand the motion accurately.