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The figure below shows a track with a small marble starting from rest at a height h, above the point where the marble leaves the track. When the marble leaves the track, it leaves at an angle e with the horizontal and at a height h, above the ground. It then flies through the air until it hits the ground at a distance D to the right from the bottom of the table. Starts on Track Here Leaves Track Here Lands on Ground Here D. A sample path taken by the marble

The figure below shows a track with a small marble starting from rest at a height h, above the point where the marble leaves the track. When the marble leaves the track, it leaves at an angle e with the horizontal and at a height h, above the ground. It then flies through the air until it hits the ground at a distance D to the right from the bottom of the table. Starts on Track Here Leaves Track Here Lands on Ground Here D. A sample path taken by the marble

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**Problem #4: Undergraduate Dynamics Problem** The figure below shows a track with a small marble starting from rest at a height \( h_t \) above the point where the marble leaves the track. When the marble leaves the track, it leaves at an angle \( \theta \) with the horizontal and at a height \( h_g \) above the ground. It then flies through the air until it hits the ground at a distance \( D \) to the right from the bottom of the table. **Diagram Description:** - The path starts at a height \( h_t \). - The trajectory shows the marble leaving the track at height \( h_g \) with an angle \(\theta\), and landing on the ground. - The horizontal distance from where the marble leaves the track to the landing point is labeled \( D \). **A sample path taken by the marble** Given the values of \( h_t \), \( h_g \) and the launch angle \(\theta\): a.) Determine an expression for \( D \). b.) Determine (in terms of \( \gamma = h_g/h_t \)) the launch angle so that \( D \) is as large as possible. c.) Determine (in terms of \( h_t \) and \( h_g \)) this largest value of \( D \) (call it \( D_{\text{max}} \)). d.) Compute these values using \( h_t = 1.5 \text{ feet}, h_g = 3.0 \text{ feet} \) and \( g = 32 \text{ feet/sec}^2 \).
Transcribed Image Text:**Problem #4: Undergraduate Dynamics Problem** The figure below shows a track with a small marble starting from rest at a height \( h_t \) above the point where the marble leaves the track. When the marble leaves the track, it leaves at an angle \( \theta \) with the horizontal and at a height \( h_g \) above the ground. It then flies through the air until it hits the ground at a distance \( D \) to the right from the bottom of the table. **Diagram Description:** - The path starts at a height \( h_t \). - The trajectory shows the marble leaving the track at height \( h_g \) with an angle \(\theta\), and landing on the ground. - The horizontal distance from where the marble leaves the track to the landing point is labeled \( D \). **A sample path taken by the marble** Given the values of \( h_t \), \( h_g \) and the launch angle \(\theta\): a.) Determine an expression for \( D \). b.) Determine (in terms of \( \gamma = h_g/h_t \)) the launch angle so that \( D \) is as large as possible. c.) Determine (in terms of \( h_t \) and \( h_g \)) this largest value of \( D \) (call it \( D_{\text{max}} \)). d.) Compute these values using \( h_t = 1.5 \text{ feet}, h_g = 3.0 \text{ feet} \) and \( g = 32 \text{ feet/sec}^2 \).
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