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A cart, of mass m, is released at the top of a ramp of distance d and height h as shown. Neglect backwards friction and drag. Calculate the final speed (v) in each of the following cases. m [kg] h [m] d[m] V [m/s] 3 0.5 1.5 6 3 3 6 0.5 1.00 0.5 1.00 1.5 1.5 3.00 3.00 h reference level

A cart, of mass m, is released at the top of a ramp of distance d and height h as shown. Neglect backwards friction and drag. Calculate the final speed (v) in each of the following cases. m [kg] h [m] d[m] V [m/s] 3 0.5 1.5 6 3 3 6 0.5 1.00 0.5 1.00 1.5 1.5 3.00 3.00 h reference level

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**Educational Content: Energy on an Inclined Plane** A cart, of mass \( m \), is released at the top of a ramp with distance \( d \) and height \( h \) as illustrated. Assume that friction and drag are negligible. **Objective:** Calculate the final speed (\( v_f \)) of the cart in each scenario provided. --- **Table of Values:** | \( m \) [kg] | \( h \) [m] | \( d \) [m] | \( v_f \) [m/s] | |--------------|-------------|-------------|-----------------| | 3 | 0.5 | 1.5 | [ ] | | 6 | 0.5 | 1.5 | [ ] | | 3 | 1.0 | 1.5 | [ ] | | 3 | 0.5 | 3.0 | [ ] | | 6 | 1.0 | 3.0 | [ ] | --- **Diagram Explanation:** - The diagram shows a ramp inclined at an angle with a cart at the top and bottom of the incline. - Distance \( d \) represents the length of the ramp from the top to the bottom. - Height \( h \) indicates the vertical drop from the top to the bottom of the ramp. - A “reference level” is marked at the bottom of the ramp. --- **Conceptual Understanding:** The potential energy at the top of the ramp transforms into kinetic energy at the bottom. Use the following energy conservation formula for calculation: \[ mgh = \frac{1}{2}mv_f^2 \] Where: - \( m \) is the mass of the cart. - \( g \) is the acceleration due to gravity (approx. \( 9.81 \, \text{m/s}^2 \)). - \( h \) is the height of the ramp. - \( v_f \) is the final speed of the cart at the bottom. Solve for \( v_f \) to find the final speed of the cart using the given values.
Transcribed Image Text:**Educational Content: Energy on an Inclined Plane** A cart, of mass \( m \), is released at the top of a ramp with distance \( d \) and height \( h \) as illustrated. Assume that friction and drag are negligible. **Objective:** Calculate the final speed (\( v_f \)) of the cart in each scenario provided. --- **Table of Values:** | \( m \) [kg] | \( h \) [m] | \( d \) [m] | \( v_f \) [m/s] | |--------------|-------------|-------------|-----------------| | 3 | 0.5 | 1.5 | [ ] | | 6 | 0.5 | 1.5 | [ ] | | 3 | 1.0 | 1.5 | [ ] | | 3 | 0.5 | 3.0 | [ ] | | 6 | 1.0 | 3.0 | [ ] | --- **Diagram Explanation:** - The diagram shows a ramp inclined at an angle with a cart at the top and bottom of the incline. - Distance \( d \) represents the length of the ramp from the top to the bottom. - Height \( h \) indicates the vertical drop from the top to the bottom of the ramp. - A “reference level” is marked at the bottom of the ramp. --- **Conceptual Understanding:** The potential energy at the top of the ramp transforms into kinetic energy at the bottom. Use the following energy conservation formula for calculation: \[ mgh = \frac{1}{2}mv_f^2 \] Where: - \( m \) is the mass of the cart. - \( g \) is the acceleration due to gravity (approx. \( 9.81 \, \text{m/s}^2 \)). - \( h \) is the height of the ramp. - \( v_f \) is the final speed of the cart at the bottom. Solve for \( v_f \) to find the final speed of the cart using the given values.
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