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Chapter 3: Problem 68

Tossing Your Lunch. Henrietta is going off to her physics class, jogging down the sidewalk at 3.05 \(\mathrm{m} / \mathrm{s}\) . Her husband Bruce suddenly realizes that she left in such a hurry that she forgot her lunch of bagels, so he runs to the window of their apartment, which is 43.9 \(\mathrm{m}\) above the street level and directly above the sidewalk, to throw them to her. Bruce throws them horizontally 9.00 \(\mathrm{s}\) after Henrietta has passed below the window, and she catches them on the run. You can ignore air resistance. (a) With what initial speed must Bruce throw the bagels so Henrietta can catch them just before they hit the ground? (b) Where is Henrietta when she catches the bagels?

Short Answer

Expert verified
(a) 12.23 m/s, (b) 36.57 m from the window.

Step by step solution

01

Define the coordinate system

Let's set the origin of our coordinate system at the location where Bruce is throwing the bagels. The vertical position of the bagels is positive downward, with initial position 0 at the window. Horizontal motion is positive in the direction of Henrietta's travel.
02

Determine time of flight

Since Bruce is throwing the bagels horizontally, their vertical motion is solely influenced by gravity. Use the equation for vertical motion under gravity to find the time it takes to hit the ground: \[ y = \frac{1}{2} g t^2 \].Here, \( y = 43.9 \ m \) and \( g = 9.81 \ m/s^2 \). Solve for \( t \):\[ 43.9 = \frac{1}{2} \times 9.81 \times t^2 \] \[ t^2 = \frac{43.9 \times 2}{9.81} = 8.949 \]\[ t \approx 2.99 \ s \]. This is the time of flight of the bagels.
03

Calculate horizontal distance traveled by the bagels

Since horizontal velocity \( v_x \) is constant, the horizontal distance \( x \) can be calculated by:\[ x = v_x \cdot t \].
04

Determine when Henrietta reaches the catching point

Henrietta started her motion 9.00 seconds before Bruce threw the bagels. So, her travel time to the point of catching the bagels is:\[ t_{Henrietta} = 9.00 + 2.99 = 11.99 \ s \].
05

Calculate horizontal position of Henrietta

Henrietta jogs at 3.05 \( m/s \). The distance she covers in \( 11.99 \ s \) is:\[ d_{Henrietta} = 3.05 \ m/s \times 11.99 \ s = 36.57 \ m \].This is the position where she catches the bagels.
06

Equate horizontal distances for both

For Henrietta to catch the bagels, the horizontal distance covered by the bagels must equal the distance covered by Henrietta:\[ v_x \times 2.99 = 36.57 \].Solve for \( v_x \):\[ v_x = \frac{36.57}{2.99} \approx 12.23 \ m/s \].

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Key Concepts

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

Horizontal Motion
In projectile motion, horizontal motion refers to the movement of an object parallel to the horizontal axis. This motion occurs at a constant velocity, meaning there is no acceleration if air resistance is neglected. Bruce's task is to throw the bagels horizontally from a high window to Henrietta below. Here, the horizontal velocity is crucial as it determines how far the bagels will travel horizontally by the time Henrietta catches them.

Key points to note:
  • The horizontal velocity (\( v_x \)) remains unchanged throughout the motion.
  • Horizontal distance (\( x \)) traveled can be calculated by multiplying velocity by time (\( x = v_x \cdot t \)).
  • This constancy is due to the absence of horizontal forces such as friction or wind resistance in this problem.
To ensure Henrietta catches the bagels just before they touch the ground, Bruce must throw them with an initial horizontal speed that allows them to cover the distance Henrietta will run in the bagels' flight duration. In this scenario, with a time of flight of approximately 2.99 seconds, the calculated horizontal velocity for the bagels to match Henrietta's position is 12.23 m/s.
Vertical Motion
The vertical motion of an object in projectile motion is governed by the force of gravity. Unlike horizontal motion, vertical motion involves acceleration. Gravity causes objects to accelerate downward at a constant rate of approximately 9.81 m/s虏. In this context, the bagels' vertical position changes from the window height down to Henrietta on the sidewalk.

Highlights of vertical motion:
  • The vertical distance (\( y \)) is influenced by the gravitational acceleration (\( g \)).
  • The vertical motion equation, \( y = \frac{1}{2} g t^2 \), helps determine the total time objects take to fall a certain height.
  • This time is solely dependent on the initial vertical height and gravitational acceleration.
Using the example, the bagels are thrown from a height of 43.9 meters. By applying the equation for free-falling objects, the time it takes for the bagels to reach the ground is approximately 2.99 seconds. This factor helps us understand how long Henrietta has until she needs to reach the spot to catch the bagels.
Kinematics
Kinematics is the branch of physics that describes the motion of objects without considering the forces that cause the motion. It involves parameters like displacement, velocity, and acceleration. In this problem, understanding kinematics is necessary to determine both the horizontal and vertical components of the bagels' motion.

Essential elements of kinematics in this scenario:
  • Decomposing motion into horizontal and vertical components.
  • Utilizing equations of motion for each component separately.
  • Combining results to predict the overall trajectory and ensuring successful interception.
With kinematics, you leverage distinct equations to solve for time, distance, and velocity, depending on the direction of motion. By separately analyzing horizontal and vertical motions, you can predict the bagels' pathway and timing. This is crucial for Bruce and Henrietta to synchronize the throw and catch, ensuring the bagels land safely in Henrietta's hands without hitting the ground prematurely.

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Most popular questions from this chapter

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