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Chapter 6: Problem 4

A child throws three identical marbles from the same height above the ground so that they land on the flat roof of a building. The marbles are launched with the same initial speed. The first marble, marble \(\mathrm{A}\), is thrown at an angle of \(75^{\circ}\) above horizontal, while marbles \(\mathrm{B}\) and \(\mathrm{C}\) are thrown with launch angles of \(60^{\circ}\) and \(45^{\circ}\), respectively. Neglecting air resistance, rank the marbles according to the speeds with which they hit the roof. a) \(A

Short Answer

Expert verified
Question: Rank the speeds of marbles A, B, and C when they hit the roof if they are thrown with the same initial speed but different angles (A: 75°, B: 60°, C: 45°) from the ground. Answer: A<B<C

Step by step solution

01

Initial horizontal velocities of each marble

To start, we need to find the initial horizontal velocities of each marble. We know that all marbles are thrown with the same initial speed, so we can represent that with a variable, such as \(v_0\). For each marble, we need to find the horizontal component of the velocity, which can be calculated using the formulas: \(v_{A_x} = v_0 \cos{75^\circ}\) \(v_{B_x} = v_0 \cos{60^\circ}\) \(v_{C_x} = v_0 \cos{45^\circ}\)
02

Calculating the initial horizontal velocities

Using a calculator, we can find the cosine of each angle: \(\cos{75^\circ} \approx 0.2588\) \(\cos{60^\circ} = 0.5\) \(\cos{45^\circ} \approx 0.7071\) Now, we can use these values to find the initial horizontal velocities: \(v_{A_x} = v_0 \times 0.2588\) \(v_{B_x} = v_0 \times 0.5\) \(v_{C_x} = v_0 \times 0.7071\)
03

Comparing horizontal velocities

The initial horizontal velocities can be ranked as: \(v_{A_x} < v_{B_x} < v_{C_x}\) As the marbles are thrown horizontally, we can assume that the horizontal velocities do not change during the process (air resistance is neglected). So, the above ranking represents the speeds at which marbles hit the roof.
04

Write the final answer

Using the ranking found in Step 3, we can write the final answer as: A is slower than B, and B is slower than C, which can be represented as: \(A<B<C\) So, the correct answer for this exercise is option (a).

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