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Chapter 7: Problem 23

A soccer ball with a mass of \(442 \mathrm{~g}\) bounces off the crossbar of a goal and is deflected upward at an angle of \(58.0^{\circ}\) with respect to horizontal. Immediately after the deflection, the kinetic energy of the ball is \(49.5 \mathrm{~J} .\) What are the vertical and horizontal components of the ball's momentum immediately after striking the crossbar?

Short Answer

Expert verified
Answer: The vertical and horizontal components of the ball's momentum are approximately 3.32 kg⋅m/s and 5.33 kg⋅m/s, respectively.

Step by step solution

01

Calculate the initial velocity of the ball

First, we'll use the given kinetic energy to find the initial velocity of the ball after the deflection. The formula to calculate kinetic energy is: \(KE = \frac{1}{2}mv^2\) We know that the kinetic energy (\(KE\)) is \(49.5 J\), the mass (\(m\)) is \(442 g = 0.442 kg\) and we need to find the velocity (\(v\)). We can rewrite the formula to solve for \(v\): \(v = \sqrt{\frac{2 \times KE}{m}}\)
02

Plug in the values and compute the initial velocity

Now, we plug in the values of \(KE\) and \(m\) into the formula: \(v = \sqrt{\frac{2 \times 49.5 J}{0.442 kg}}\) \(v \approx 14.05\,\text{m/s}\)
03

Calculate the vertical and horizontal components of the velocity

Now that we have the initial velocity, we can compute the vertical and horizontal components of the velocity using the deflection angle (\(58.0^\circ\)). We use trigonometry to find these components: \(v_x = v\cos(58.0^\circ)\) \(v_y = v\sin(58.0^\circ)\)
04

Plug in the values and compute the velocity components

We'll now plug in the values of \(v\) and the angle to compute \(v_x\) and \(v_y\): \(v_x = 14.05\,\text{m/s} \times \cos(58.0^\circ) \approx 7.52\,\text{m/s}\) \(v_y = 14.05\,\text{m/s} \times \sin(58.0^\circ) \approx 12.05\,\text{m/s}\)
05

Calculate the vertical and horizontal components of the momentum

Finally, we can find the vertical and horizontal components of the momentum by multiplying the respective components of the velocity by the mass of the ball: \(p_x = m \times v_x\) \(p_y = m \times v_y\)
06

Plug in the values and compute the momentum components

Now, we'll plug in the values of \(m\), \(v_x\), and \(v_y\) to compute \(p_x\) and \(p_y\): \(p_x = 0.442 kg \times 7.52\,\text{m/s} \approx 3.32\,\text{kg}\cdot\text{m/s}\) \(p_y = 0.442 kg \times 12.05\,\text{m/s} \approx 5.33\,\text{kg}\cdot\text{m/s}\) The vertical and horizontal components of the ball's momentum immediately after striking the crossbar are approximately \(3.32\,\text{kg}\cdot\text{m/s}\) and \(5.33\,\text{kg}\cdot\text{m/s}\), respectively.

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Most popular questions from this chapter

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