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Chapter 3: Problem 64

Show that for a projectile launched at an angle of \(45^{\circ}\) the maximum height of the projectile is one quarter of the range (the distance traveled on flat ground).

Short Answer

Expert verified
Answer: When a projectile is launched at a 45-degree angle, its maximum height is one quarter of its range.

Step by step solution

01

Define the initial conditions

Given a launch angle of \(45^{\circ}\), we can write the initial velocity components as follows: - Initial horizontal velocity: \(v_{0x} = v_0 \cos 45^{\circ} = \frac{v_0}{\sqrt{2}}\) - Initial vertical velocity: \(v_{0y} = v_0 \sin 45^{\circ} = \frac{v_0}{\sqrt{2}}\)
02

Obtain expressions for the maximum height and range

To find the expressions for maximum height (\(h_{max}\)) and range (\(R\)), we'll use the equations for displacement with uniform acceleration: - Maximum height: At maximum height, the vertical velocity becomes zero, i.e., \(v_y = 0\). We will use the following equation: \(v_y^2 = v_{0y}^2 - 2gh_{max}\) - Range: To calculate the range, we need to find the time of flight (\(t_f\)) and then use it to calculate the horizontal displacement. The time of flight can be found using the vertical motion equation: \(v_y = v_{0y} - gt_f\)
03

Calculate the maximum height

At maximum height, \(v_y = 0\). So, using the equation from step 2 and replacing \(v_{0y}\) with its value from step 1, we get: \(0 = (\frac{v_0}{\sqrt{2}})^2 - 2g h_{max}\) Solve for \(h_{max}\): \(h_{max} = \frac{v_0^2}{4g}\)
04

Calculate the range

Using the vertical motion equation from step 2 and replacing \(v_{0y}\) and \(v_y\) with their values from step 1, we get: \(0 = \frac{v_0}{\sqrt{2}} - g t_f\) Now, solve for \(t_f\): \(t_f = \frac{v_0}{\sqrt{2}g}\) The horizontal displacement equation is: \(R = v_{0x} t_f\) Using values of \(v_{0x}\) and \(t_f\) from steps 1 and 4: \(R = \frac{v_0}{\sqrt{2}} \cdot \frac{v_0}{\sqrt{2}g} = \frac{v_0^2}{2g}\)
05

Prove that the maximum height is one quarter of the range

Now, let's compare the expressions for \(h_{max}\) and \(R\): \(\frac{h_{max}}{R} = \frac{\frac{v_0^2}{4g}}{\frac{v_0^2}{2g}} = \frac{1}{2}\) Thus, the maximum height is one quarter of the range when the launch angle is \(45^{\circ}\).

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Most popular questions from this chapter

In each of these, the \(x\) - and \(y\) -components of a vector are given. Find the magnitude and direction of the vector. (a) $A_{x}=-5.0 \mathrm{m} / \mathrm{s}, A_{y}=+8.0 \mathrm{m} / \mathrm{s} .\( (b) \)B_{x}=+120 \mathrm{m}$ $B_{y}=-60.0 \mathrm{m} .(\mathrm{c}) C_{x}=-13.7 \mathrm{m} / \mathrm{s}, C_{y}=-8.8 \mathrm{m} / \mathrm{s} .(\mathrm{d}) D_{x}=$ \(2.3 \mathrm{m} / \mathrm{s}^{2}, D_{y}=6.5 \mathrm{cm} / \mathrm{s}^{2}\)
A small plane is flying directly west with an airspeed of $30.0 \mathrm{m} / \mathrm{s} .\( The plane flies into a region where the wind is blowing at \)10.0 \mathrm{m} / \mathrm{s}\( at an angle of \)30^{\circ}$ to the south of west. In that region, the pilot changes the directional heading to maintain her due west heading. (a) What is the change she makes in the directional heading to compensate for the wind? (b) After the heading change, what is the ground speed of the airplane?
A suspension bridge is \(60.0 \mathrm{m}\) above the level base of a gorge. A stone is thrown or dropped from the bridge. Ignore air resistance. At the location of the bridge \(g\) has been measured to be $9.83 \mathrm{m} / \mathrm{s}^{2} .$ (a) If you drop the stone, how long does it take for it to fall to the base of the gorge? (b) If you throw the stone straight down with a speed of \(20.0 \mathrm{m} / \mathrm{s},\) how long before it hits the ground? (c) If you throw the stone with a velocity of \(20.0 \mathrm{m} / \mathrm{s}\) at \(30.0^{\circ}\) above the horizontal, how far from the point directly below the bridge will it hit the level ground?
Vector \(\overrightarrow{\mathbf{A}}\) is directed along the positive \(y\) -axis and has magnitude \(\sqrt{3.0}\) units. Vector \(\overrightarrow{\mathbf{B}}\) is directed along the negative \(x\) -axis and has magnitude 1.0 unit. (a) What are the magnitude and direction of \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}} ?\) (b) What are the magnitude and direction of \(\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}} ?\) (c) What are the \(x\) - and \(y\) -components of \(\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}} ?\)
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