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Chapter 2: Problem 4

A clown is juggling four balls simultaneously. Students use a video tape to determine that it takes the clown 0.9 s to cycle each ball through his hands (including catching, transferring, and throwing) and to be ready to catch the next ball. What is the minimum vertical speed the clown must throw up each ball?

Short Answer

Expert verified
The minimum vertical speed the clown must throw up each ball is approximately 1.103 m/s.

Step by step solution

01

Identify the known variables and equations

We are given the following information: - Time taken for one cycle = 0.9 seconds - Number of balls = 4 - Acceleration due to gravity (g) = 9.81 m/s虏 (approximately) We will use the equation of motion for vertical motion under constant acceleration (due to gravity): \(y = v_0t + \frac{1}{2}gt^2\), where - y is the vertical displacement (0 for our problem since the balls will be back to the same height at the end of each cycle) - \(v_0\) is the initial vertical speed (what we want to determine) - t is the time - g is the acceleration due to gravity
02

Calculate the time for each ball to reach the highest point

Since the clown juggles 4 balls, the time taken for each ball \(t' = \frac{t}{4}\) So, calculate \(t'\): \(t' = \frac{0.9}{4}\) \(t' = 0.225\) seconds
03

Use the equation of motion to find initial vertical speed

To find the minimum initial vertical speed, we can use the equation of motion and the condition that vertical displacement y = 0 at time t': \(0 = v_0t' - \frac{1}{2}gt'^2\) Rearrange the equation to get: \(v_0 = \frac{1}{2}gt'\) Now plug in the values for g and \(t'\) and calculate the initial vertical speed: \(v_0 = \frac{1}{2}(9.81)(0.225)\) \(v_0 \approx 1.103\) m/s Therefore, the clown must throw up each ball with a minimum vertical speed of approximately 1.103 m/s.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Projectile Motion
Projectile motion involves objects that are launched into the air and are influenced only by gravity. These objects, like a ball being juggled, follow a curved path called a trajectory.
The path is determined by two components: horizontal and vertical motion.
  • In this clown exercise, each ball moves upward and then downward, staying in motion due to the initial force applied by the clown.
  • Gravity pulls the balls back down, affecting their vertical motion.
No horizontal force acts on the balls, so they move horizontally at a constant velocity. Understanding projectile motion means analyzing what happens in both directions separately but simultaneously.
Equations of Motion
Equations of motion help to predict the future positions and velocities of objects in motion.
In projectile motion, we use these equations to calculate how high and how far the objects will travel. The key equation in this problem is for vertical displacement:
  • Vertical displacement \( y = v_0t + \frac{1}{2}gt^2 \)
  • \( v_0 \) is the initial speed.
  • \( g \) is the acceleration due to gravity.
  • \( t \) is the time taken.
This equation allows us to solve for unknowns, like finding the initial vertical speed required for the juggled balls.
By calculating how long each ball stays in the air, we can ensure they return to the clown's hands in time.
Vertical Speed Calculation
Calculating the vertical speed needed requires using the equation of motion for vertical displacement with some clever rearrangement. We know the ball makes complete round trips in the air, returning to the clown's hand exactly every 0.9 seconds.
By splitting this cycle among the four balls, each ball has 0.225 seconds to reach its highest point.
Key insights:
  • At the peak, the vertical speed is zero because the ball stops climbing and starts descending.
  • Using the equation \( 0 = v_0t' - \frac{1}{2}gt'^2 \), we rearrange to find \( v_0 \).
  • Substituting \( t' = 0.225 \) seconds and \( g = 9.81 \text{ m/s}^2 \), we find \( v_0 \approx 1.103 \text{ m/s} \).
This calculation guarantees the juggling act proceeds without a hitch, ensuring the clown can catch, transfer, and throw each ball on time.

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Most popular questions from this chapter

Consider a projectile fired vertically in a constant gravitational field. For the same initial velocities, compare the times required for the projectile to reach its maximum height (a) for zero resisting force, (b) for a resisting force proportional to the instantaneous velocity of the projectile.

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The speed of a particle of mass \(m\) varies with the distance \(x\) as \(v(x)=\alpha x^{-n}\). Assume \(v(x=0)=0\) at \(t=0 .\) (a) Find the force \(F(x)\) responsible. (b) Determine \(x(t)\) and (c) \(F(t)\).

A particle is under the influence of a force \(F=-k x+k x^{3} / \alpha^{2},\) where \(k\) and \(\alpha\) are constants and \(k\) is positive. Determine \(U(x)\) and discuss the motion. What happens when \(E=(1 / 4) k \alpha^{2} ?\)

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