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Chapter 2: Problem 63

What is the velocity at the midway point of a ball able to reach a height \(y\) when thrown with an initial velocity \(v_{0} ?\)

Short Answer

Expert verified
Answer: The velocity at the midway point of the ball is \(\frac{v_{0}}{2}\).

Step by step solution

01

Identify the equations of motion

We will use the following equations of motion to solve the problem: 1. \(v=v_{0}+at\) 2. \(y=v_{0}t+\frac{1}{2}at^2\) Here, \(v\) is the final velocity of the ball, \(v_{0}\) is the initial velocity, \(a\) is the acceleration due to gravity, \(t\) is the time taken to reach that height y, and \(y\) is the height reached by the ball.
02

Determine the height at the midway point

Since the midway point corresponds to half the maximum height of the ball, we will consider the height \(y\) to be \(\frac{1}{2}y_{max}\). So let's denote it as: \(h = \frac{1}{2}y_{max}\)
03

Calculate the maximum height reached

To find the maximum height reached by the ball, we can use the equation of motion when the ball's velocity at the maximum height becomes 0: \(0^2 = v_{0}^2 - 2ay_{max}\) Solving for \(y_{max}\), we get: \(y_{max} = \frac{v_{0}^2}{2a}\), where \(a = -g\), the acceleration due to gravity. (We consider it negative since it's acting opposite to the direction of the throw)
04

Determine the time taken to reach midway point

To find the time taken to reach the midway point, we will use the height at the midway point (h) and the second equation of motion: \(h = v_{0}t+\frac{1}{2}at^2\) Substitute \(h\) with \(\frac{1}{2}y_{max}\) and \(y_{max}\) with \(\frac{v_{0}^2}{2a}\): \(\frac{1}{2}\times\frac{v_{0}^2}{2a} = v_{0}t+\frac{1}{2}at^2\) Solving for t, we get: \(t = \frac{v_{0}}{2a}\)
05

Calculate the velocity at the midway point

Now that we have the time taken to reach the midway point, we can find the velocity at this point using the first equation of motion. \(v = v_{0} + at\) Plug in the values of \(t\) and \(a\): \(v = v_{0} - g\frac{v_{0}}{2g}\) \(v = \frac{v_{0}}{2}\) So, the velocity at the midway point of a ball able to reach a height \(y\) when thrown with an initial velocity \(v_{0}\) is \(\frac{v_{0}}{2}\).

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Most popular questions from this chapter

In a fancy hotel, the back of the elevator is made of glass so that you can enjoy a lovely view on your ride. The elevator travels at an average speed of \(1.75 \mathrm{~m} / \mathrm{s}\). A boy on the 15th floor, \(80.0 \mathrm{~m}\) above the ground level, drops a rock at the same instant the elevator starts its ascent from the 1st to the 5th floor. Assume the elevator travels at its average speed for the entire trip and neglect the dimensions of the elevator. a) How long after it was dropped do you see the rock? b) How long does it take for the rock to reach ground level?

A bullet is fired through a board \(10.0 \mathrm{~cm}\) thick, with a line of motion perpendicular to the face of the board. If the bullet enters with a speed of \(400 . \mathrm{m} / \mathrm{s}\) and emerges with a speed of \(200 . \mathrm{m} / \mathrm{s}\), what is its acceleration as it passes through the board?

An F-14 Tomcat fighter jet is taking off from the deck of the USS Nimitz aircraft carrier with the assistance of a steam-powered catapult. The jet's location along the flight deck is measured at intervals of \(0.20 \mathrm{~s} .\) These measurements are tabulated as follows: $$ \begin{array}{|l|l|l|l|l|l|l|l|l|l|l|l|} \hline t(\mathrm{~s}) & 0.00 & 0.20 & 0.40 & 0.60 & 0.80 & 1.00 & 1.20 & 1.40 & 1.60 & 1.80 & 2.00 \\ \hline x(\mathrm{~m}) & 0.0 & 0.70 & 3.0 & 6.6 & 11.8 & 18.5 & 26.6 & 36.2 & 47.3 & 59.9 & 73.9 \\ \hline \end{array} $$ Use difference formulas to calculate the jet's average velocity and average acceleration for each time interval. After completing this analysis, can you say if the F- 14 Tomcat accelerated with approximately constant acceleration?

A fighter jet lands on the deck of an aircraft carrier. It touches down with a speed of \(70.4 \mathrm{~m} / \mathrm{s}\) and comes to a complete stop over a distance of \(197.4 \mathrm{~m}\). If this process happens with constant deceleration, what is the speed of the jet \(44.2 \mathrm{~m}\) before its final stopping location?

Which of these statement(s) is (are) true? 1\. An object can have zero acceleration and be at rest. 2\. An object can have nonzero acceleration and be at rest. 3\. An object can have zero acceleration and be in motion. a) 1 only b) 1 and 3 c) 1 and 2 d) \(1,2,\) and 3
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