Q16. Two players collide: A 70.0 kg running back (north, 5.00 m/s) and an 80.0 kg linebacker (west, 4.00 m/s) grab on. (a) Find x and y components of total momentum before collision. (b) Find the velocity (magnitude and direction) of the pair just after. (c) How much KE is lost?

Background

Topic: Conservation of Momentum & Kinetic Energy in Collisions

This question tests your understanding of vector momentum, conservation of momentum in two dimensions, and energy loss in inelastic collisions.

Key Terms and Formulas:

• Momentum:

• Conservation of Momentum:

• Kinetic Energy:

Step-by-Step Guidance

1. Assign coordinate axes: Let north be the direction and west be the direction.

2. Calculate the momentum components for each player:

   • Running back: ,

   • Linebacker: ,

3. Add the momentum components to find total and before collision.

4. After collision, the players stick together. Use conservation of momentum to find the combined velocity vector:

5. To find the magnitude and direction, use:

   • Magnitude:

   • Direction:

6. For kinetic energy lost, compare total KE before and after collision:

   • Calculate for each player and sum.

   • Calculate for the combined mass and velocity.

   • KE lost =

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Q17. A 4.00 kg bomb at rest explodes into three pieces. Piece 1 (1.00 kg) flies north at 20.0 m/s. Piece 2 (1.50 kg) flies east at 16.0 m/s. (a) Find the mass of Piece 3. (b) Find Piece 3's velocity (magnitude and direction). (c) Find total KE released.

Background

Topic: Conservation of Momentum & Energy in Explosions

This question tests your ability to apply conservation of momentum in two dimensions and calculate kinetic energy released.

Key Terms and Formulas:

• Conservation of Momentum:

• Kinetic Energy:

Step-by-Step Guidance

1. Find the mass of Piece 3:

2. Set up momentum conservation in both and directions. The bomb starts at rest, so total momentum after must be zero.

3. Calculate and for Pieces 1 and 2, then solve for Piece 3's and (they must cancel the others).

4. Find Piece 3's velocity components: ,

5. Calculate magnitude and direction: ,

6. Find total KE released: sum for all three pieces.

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Q18. Potential energy function: (J, in m). (a) Find all equilibrium positions (). (b) Determine which are stable (local minimum) and which are unstable. (c) Describe the motion of a 0.500 kg object released from rest at . (d) Find its speed at m.

Background

Topic: Potential Energy, Equilibrium, and Motion in a Potential

This question tests your understanding of equilibrium points, stability, and energy conservation.

Key Terms and Formulas:

• Equilibrium:

• Stability: (second derivative test)

• Conservation of Energy:

• Kinetic Energy:

Step-by-Step Guidance

1. Find equilibrium positions by setting and solving for .

2. Use the second derivative to determine stability at each equilibrium.

3. For , describe the motion based on the potential shape and force direction.

4. Use energy conservation to find speed at m:

5. Set up the equation for at m, but stop before plugging in numbers.

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Q19. A 0.300 kg block is dropped from 0.800 m onto a spring ( N/m) on the floor. (a) Find maximum spring compression. (b) Find maximum elastic PE stored. (c) At maximum compression, what is the net upward force on the block? (d) After bouncing, how high above the drop point does the block rise?

Background

Topic: Energy Conservation, Springs, and Forces

This question tests your understanding of gravitational potential energy, elastic potential energy, and forces at equilibrium.

Key Terms and Formulas:

• Gravitational PE:

• Elastic PE:

• Force from spring:

• Net force:

Step-by-Step Guidance

1. Set up energy conservation: to solve for maximum compression .

2. Calculate maximum elastic PE using .

3. At maximum compression, find and subtract to get net upward force.

4. After bouncing, use energy conservation to find the height above the drop point.

5. Set up the equations for each part, but stop before final calculations.

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Q20. An 8.00 g bullet at 350 m/s embeds in a 2.00 kg block on a 1.50 m-high table. (a) Find the block+bullet speed just after impact. (b) How far from the table base does the block land? (c) How much KE was lost in the collision?

Background

Topic: Conservation of Momentum, Projectile Motion, and Energy Loss

This question tests your ability to apply conservation of momentum, analyze projectile motion, and calculate energy loss in inelastic collisions.

Key Terms and Formulas:

• Momentum:

• Conservation of Momentum:

• Projectile motion:

• Kinetic Energy:

Step-by-Step Guidance

1. Convert bullet mass to kg: kg

2. Use conservation of momentum to find combined speed just after impact.

3. Calculate time to fall from 1.50 m using .

4. Find horizontal distance:

5. Calculate KE before and after collision to find KE lost.

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Q21. An 850 kg roller coaster car starts from rest at h = 30.0 m on a frictionless track with a loop (R = 7.50 m, bottom at ground) and then a second hill. (a) Find speed at the top of the loop. (b) Find normal force at top of loop. (c) Find maximum height h2 the car can reach after the loop.

Background

Topic: Conservation of Energy, Circular Motion, and Forces

This question tests your understanding of energy conservation, centripetal force, and normal force in circular motion.

Key Terms and Formulas:

• Energy Conservation:

• Centripetal Force:

• Normal Force at top:

Step-by-Step Guidance

1. Use energy conservation to find speed at the top of the loop: initial PE minus PE at top equals KE at top.

2. Calculate centripetal force required at the top of the loop.

3. Find normal force:

4. For maximum height after the loop, use energy conservation (subtract KE at top from total energy).

5. Set up equations for each part, but stop before final calculations.

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Q22. An engine (P = 3.00 kW) propels a 1500 kg car. Road resistance is constant at f = 600 N. (a) Find the car's maximum (terminal) speed. (b) At instantaneous speed 4.00 m/s, find the car's acceleration. (c) How much work does the engine do in 30.0 s while moving at 4.00 m/s?

Background

Topic: Power, Forces, and Work

This question tests your understanding of power, force, acceleration, and work in the context of a moving vehicle.

Key Terms and Formulas:

• Power:

• Terminal speed:

• Net force:

• Acceleration:

• Work:

Step-by-Step Guidance

1. Calculate terminal speed using

2. At m/s, find engine force:

3. Find net force:

4. Calculate acceleration:

5. Find work done in 30.0 s:

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Q23. Two flasks connected by a valve: Flask 1 (V1 = 2.00 L, 300 K, 3.00 atm); Flask 2 (V2 = 3.00 L, 300 K, 1.00 atm). Valve opens, gases mix at 300 K. (a) Find total moles before mixing. (b) Find final pressure. (c) Find rms speed of N2 at 300 K. (M = 0.0280 kg/mol, R = 8.314 J/mol K.)

Background

Topic: Ideal Gas Law and Kinetic Theory

This question tests your ability to use the ideal gas law and calculate molecular speeds.

Key Terms and Formulas:

• Ideal Gas Law:

• RMS speed:

Step-by-Step Guidance

1. Calculate moles in each flask using

2. Add moles for total before mixing.

3. After mixing, total volume is , total moles is sum, temperature is constant.

4. Use to find final pressure.

5. Calculate for N2 at 300 K.

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Q24. A piston compresses 0.500 mol of ideal gas from V1 = 10.0 L to V2 = 2.50 L at constant T = 350 K. (a) Find initial and final pressures. (b) The gas is then heated at constant volume until pressure triples. Find final temperature. (c) Find total work done on the gas in both processes.

Background

Topic: Ideal Gas Law, Thermodynamics, and Work

This question tests your understanding of gas laws, isothermal and isochoric processes, and work calculations.

Key Terms and Formulas:

• Ideal Gas Law:

• Isothermal work:

• Isochoric process:

Step-by-Step Guidance

1. Use to find and for initial and final volumes.

2. For constant volume heating, pressure triples:

3. Calculate work done during isothermal compression using

4. Work done during isochoric heating is zero.

5. Set up total work calculation, but stop before final computation.

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Q25. A 0.500 kg copper block (c = 390 J/kg K) at 200 C is dropped into 2.00 kg of water at 20.0 C (c_water = 4186 J/kg K) in an insulated container. (a) Find the final equilibrium temperature. (b) How much heat did the copper lose? (c) How much heat did the water gain? (d) Verify heat lost equals heat gained.

Background

Topic: Calorimetry and Heat Transfer

This question tests your understanding of heat exchange, specific heat, and energy conservation.

Key Terms and Formulas:

• Heat transfer:

• Energy conservation:

Step-by-Step Guidance

1. Set up energy balance:

2. Solve for (final equilibrium temperature).

3. Calculate heat lost by copper:

4. Calculate heat gained by water:

5. Check that

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Q26. How much heat converts 0.200 kg of ice at -20.0 C to steam at 130 C? (c_ice = 2090 J/kg K, c_water = 4186 J/kg K, c_steam = 2010 J/kg K; Lf = 3.34 x 10^5 J/kg, Lv = 2.26 x 10^6 J/kg.) Find the heat for each of five stages: (a) warm ice from -20 to 0 C, (b) melt ice, (c) warm water from 0 to 100 C, (d) boil water, (e) warm steam from 100 to 130 C, and (f) find the total.

Background

Topic: Phase Changes and Heat Calculations

This question tests your understanding of heat required for temperature changes and phase transitions.

Key Terms and Formulas:

• Heat for temperature change:

• Heat for melting:

• Heat for boiling:

Step-by-Step Guidance

1. Calculate heat to warm ice:

2. Calculate heat to melt ice:

3. Calculate heat to warm water:

4. Calculate heat to boil water:

5. Calculate heat to warm steam:

6. Add all heats for total:

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Q27. A monatomic ideal gas (Cv = 3R/2 per mole) undergoes a cycle: (1) isochoric heating from T1 = 300 K, p1 = 1.00 x 10^5 Pa, V = 0.0200 m^3 to p2 = 3.00 x 10^5 Pa; (2) isobaric expansion back to T1; (3) isothermal compression back to the original state. (a) Find T after step 1. (b) Find V after step 2. (c) Find work done by gas in each step. (d) Find net work per cycle.

Background

Topic: Thermodynamic Cycles and Work

This question tests your understanding of thermodynamic processes, gas laws, and work calculations.

Key Terms and Formulas:

• Isochoric:

• Isobaric:

• Isothermal work:

• Work for isochoric:

• Work for isobaric:

Step-by-Step Guidance

1. For step 1 (isochoric), use to find .

2. For step 2 (isobaric), use to find .

3. Calculate work for each step: isochoric (), isobaric (), isothermal ().

4. Set up net work per cycle as sum of works for all steps.

5. Stop before plugging in numbers for final net work.

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