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(POTENTIAL ENERGY CURVES) A 10 kg particle moves under the potential shown below, where U(x) has units of Joules and a is measured in meters. The potential energy is 20 J for all x 210 m. (a) What is the force on the particle when it is at x = 3 m? (b) What minimum speed (moving to the left) must the particle have at x = 11 m in order to reach x = 2 m? = (c) Describe the motion of the particle if it begins at x = 1 m with a velocity of UT +√12 m/s. A complete answer would discuss turning points and limitations to the range of motion, if any, as well as regions where the particle speed increases, decreases, and remains constant.

(POTENTIAL ENERGY CURVES) A 10 kg particle moves under the potential shown below, where U(x) has units of Joules and a is measured in meters. The potential energy is 20 J for all x 210 m. (a) What is the force on the particle when it is at x = 3 m? (b) What minimum speed (moving to the left) must the particle have at x = 11 m in order to reach x = 2 m? = (c) Describe the motion of the particle if it begins at x = 1 m with a velocity of UT +√12 m/s. A complete answer would discuss turning points and limitations to the range of motion, if any, as well as regions where the particle speed increases, decreases, and remains constant.

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**Potential Energy Curves** A 10 kg particle moves under the potential shown below, where \( U(x) \) has units of Joules and \( x \) is measured in meters. The potential energy is 20 J for all \( x \geq 10 \) m. **(a)** What is the force on the particle when it is at \( x = 3 \) m? **(b)** What minimum speed (moving to the left) must the particle have at \( x = 11 \) m in order to reach \( x = 2 \) m? **(c)** Describe the motion of the particle if it begins at \( x = 1 \) m with a velocity of \( v_x = +\sqrt{12} \) m/s. A complete answer would discuss turning points and limitations to the range of motion, if any, as well as regions where the particle speed increases, decreases, and remains constant. **(d)** If the particle instead begins at \( x = 1 \) m with a velocity \( v_x > \sqrt{12} \) m/s, does the particle ever turn around? Explain. **Graph Explanation:** - The graph displays potential energy \( U(x) \) in Joules on the y-axis and position \( x \) in meters on the x-axis. - For \( x \leq 0 \), \( U(x) \) is at 80 J. - From \( x = 0 \) to \( x = 2 \), \( U(x) \) decreases linearly down to -20 J. - From \( x = 2 \) to \( x = 4 \), \( U(x) \) rises sharply to 20 J. - From \( x = 4 \) to \( x = 8 \), \( U(x) \) remains constant at 20 J. - From \( x = 8 \) to \( x = 10 \), \( U(x) \) decreases linearly to reach 20 J again. - For \( x \geq 10 \), \( U(x) \) remains constant at 20 J.
Transcribed Image Text:**Potential Energy Curves** A 10 kg particle moves under the potential shown below, where \( U(x) \) has units of Joules and \( x \) is measured in meters. The potential energy is 20 J for all \( x \geq 10 \) m. **(a)** What is the force on the particle when it is at \( x = 3 \) m? **(b)** What minimum speed (moving to the left) must the particle have at \( x = 11 \) m in order to reach \( x = 2 \) m? **(c)** Describe the motion of the particle if it begins at \( x = 1 \) m with a velocity of \( v_x = +\sqrt{12} \) m/s. A complete answer would discuss turning points and limitations to the range of motion, if any, as well as regions where the particle speed increases, decreases, and remains constant. **(d)** If the particle instead begins at \( x = 1 \) m with a velocity \( v_x > \sqrt{12} \) m/s, does the particle ever turn around? Explain. **Graph Explanation:** - The graph displays potential energy \( U(x) \) in Joules on the y-axis and position \( x \) in meters on the x-axis. - For \( x \leq 0 \), \( U(x) \) is at 80 J. - From \( x = 0 \) to \( x = 2 \), \( U(x) \) decreases linearly down to -20 J. - From \( x = 2 \) to \( x = 4 \), \( U(x) \) rises sharply to 20 J. - From \( x = 4 \) to \( x = 8 \), \( U(x) \) remains constant at 20 J. - From \( x = 8 \) to \( x = 10 \), \( U(x) \) decreases linearly to reach 20 J again. - For \( x \geq 10 \), \( U(x) \) remains constant at 20 J.
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