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Chapter 3: Problem 10

A baseball is launched from the bat at an angle \(\theta_{0}=30^{\circ}\) with respect to the positive \(x\) -axis and with an initial speed of \(40 \mathrm{~m} / \mathrm{s}\), and it is caught at the same height from which it was hit. Assuming ideal projectile motion (positive \(y\) -axis upward), the velocity of the ball when it is caught is a) \((20.00 \hat{x}+34.64 \hat{y}) \mathrm{m} / \mathrm{s}\). b) \((-20.00 \hat{x}+34.64 \hat{y}) \mathrm{m} / \mathrm{s}\) c) \((34.64 \hat{x}-20.00 \hat{y}) \mathrm{m} / \mathrm{s}\) d) \((34.64 \hat{x}+20.00 \hat{y}) \mathrm{m} / \mathrm{s}\).

Short Answer

Expert verified
Short Answer: To solve this problem, we can follow these steps: 1. Calculate the initial velocity components in the x and y directions using trigonometry. 2. Calculate the time of flight when the ball reaches the same height as it was launched. 3. Calculate the final velocity components in x and y directions using equations of motion. 4. Compare the calculated components with the given options to find the correct answer. The final velocity of the baseball will be given by the components that match one of the given options.

Step by step solution

01

Calculate the Initial Velocity Components

We need to calculate the initial velocity components along the x and y direction. Since the initial speed is 40 m/s and the launch angle is \(30^{\circ}\), we can find the components using trigonometry: \(V_{0x} = V_0 \cos\theta_{0} = 40 \cos30^{\circ}\) \(V_{0y} = V_0 \sin\theta_{0} = 40 \sin30^{\circ}\)
02

Calculate the Time of Flight

To find the time of flight, we need to find out when the baseball reaches the same height it was launched from. Since there is zero initial displacement along the y-axis, we can use the following equation to find the time of flight: \(y = V_{0y}t - \frac{1}{2}gt^2\) Since the baseball is caught at the same height, \(y = 0\). Plugging the values into the above equation, we get: \(0 = V_{0y}t - \frac{1}{2}(9.8)t^2\) Solving for t, we get two possible values for the time of flight. One value will be for the time it takes to reach its peak, and the other for when it lands back to its initial height. We are interested in the landing time.
03

Calculate Final Velocity Components

We can calculate the final x and y components of the velocity by using the equations of motion: In the x-direction, the velocity remains constant: \(V_{fx} = V_{0x}\) In the y-direction, we need to account for the acceleration due to gravity: \(V_{fy} = V_{0y} - gt\) Now, we can plug in the values we have calculated earlier and find the final velocity components.
04

Compare Results with the Given Options

Now that we have calculated the final velocity components along x and y directions, we can compare them with the given options. We will find that the components match one of the given options, which will be our answer. In this way, by calculating initial velocity components, time of flight, and final velocity components, we can find the correct option for the velocity of the ball when it is caught at the same height it was hit.

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Most popular questions from this chapter

A ball is thrown straight up by a passenger in a train that is moving with a constant velocity. Where would the ball land-back in his hands, in front of him, or behind him? Does your answer change if the train is accelerating in the forward direction? If yes, how?

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