AP Practice Test – Work, Energy, Power, Momentum, Systems

1. A massless spring with force constant k = 400 N/m is fastened at its left end to a vertical wall, as shown in Figure I. Initially, block C (mass m_C = 4.0 kg) and block D (mass m_D = 2.0 kg) rest on a horizontal surface with block C in contact with the spring (but not compressing it) and with block D in contact with block C. Block C is then moved to the left compressing the spring a distance of 0.50 meter, and held in place while block D remains at rest as shown in Figure II.
(a) Determine the elastic energy stored in the compressed spring.
Block C is then released and accelerates to the right, toward block D. The surface is rough and the coefficient of friction between each block and the surface is m = 0.4. The two blocks collide instantaneously, stick together, and move to the right. Remember that the spring is not attached to block C. Determine each of the following.
(b) The speed v_C of block C just before it collides with block D.
(c) The speed v_f of blocks C and D just after they collide.
(d) The horizontal distance the blocks move before coming to rest.

2. A spring of negligible mass and constant k = 100 N/m is attached to a support at its top end and its bottom end is free to move. (Note: the spring can be stretched or compressed.) Let the bottom end of the spring be a position y = 0. The free end of the spring is stretched downward to a position y = -80.0 cm and a 3.00 kg mass is securely attached there. Initially at rest, the mass is released from this position.
(a) Determine the maximum position the mass will move upward.
(b) Determine the speed of the mass at position y = -10.0 cm.
(c) Determine the maximum speed of the mass after its release.

3. An object moving on the x-axis is subjected to a rightward force given by the function F(t) = 8t - 3t², where F is in newtons and t is in seconds. At t = 1 s, the object has momentum p = 4 kg m/s, 180°. Determine the object’s momentum at t = 2.5 s.

4. An object moving on the x-axis has momentum given by p(t) = (3 N/s)t² - (5 N/s²)t³. Determine the force acting on the object at t = 2 seconds.

5. Determine the location of the center of mass of the Earth-Moon system. The separation from center to center is 3.84 ´ 10⁸ m. Mass and radii of these bodies: Earth – 5.974 ´ 10²⁴ kg, 6378 km; Moon – 7.36 ´ 10²² kg, 1740 km. The motion of the Earth and Moon may be approximated by assuming that each body moves in a circular path about the center of mass. Determine the speed of each object in its orbit about the other (both values relative to the center of mass).

6. The Earth-Moon system is in orbit about the Sun (mass 1.99 ´ 10³⁰ kg). Determine the acceleration of the Earth-Moon system’s center of mass due to the external force of the Sun’s gravity. To simplify this calculation, assume each object is a distance from the Sun equal to 1.496 ´ 10¹¹ m (the average orbital radius of the Earth). Assuming the center of mass follows a circular path about the Sun determine the speed of the center of mass relative to the Sun.

7. Combine the results of the previous problems and determine the minimum and maximum speed of the Earth relative to the Sun. Repeat for the Moon.

8. Mercury (3.30 ´ 10²³ kg) orbits the Sun (1.99 ´ 10³⁰ kg) at a mean distance of 58.3 Gm but its elliptical orbit takes it as far as 70.4 Gm and as close as 46.3 Gm from the Sun. When it is at its mean distance its speed is 47.9 km/s. Determine its minimum and maximum speed as it orbits the Sun.

9. An 800 gram arrow is launched with velocity 40.0 m/s, 30.0° from position x = 0. At the same time a 400 gram target is launched with velocity 20.0 m/s, 90.0° from a position of x = 60.0 m. Both objects are launched from ground level. The arrow pierces the target and sticks in it. Determine where the arrow/target (single object) will land assuming the ground is level. And find the impact velocity. Hint: determine the motion of the center of mass!

10. A certain elevator car has mass 2500 kg and is lifted by cables driven by an electric motor. The efficiency of the lift system (motor, pulleys, cable, etc.) is 70.0%. As the elevator car moves at constant speed of 2.0 m/s it encounters 160 N of friction coming from its tracks. Determine the electrical power requirement for this system as it lifts the car.

11. An object with mass 2.0 kg is moving on the x-axis is subjected to a net force in newtons given by F(x) = 5 - 6x², where x is measured in meters. (a) Determine the potential energy function U(x) associated with this force; let x = 0 be the reference point. Suppose the object is initially at x = 1.0 m with speed v = 2.0 m/s. (b) Determine the farthest right and left positions to which it will move. (c) Determine the speed required at the same initial position that would be sufficient to send the object away from the origin, never to return.

Answers:

1. a. 50.0 J

b. 4.59 m/s

c. 3.06 m/s

d. 1.20 m

2. a. y = 0.212 m

b. 2.70 m/s

c. 2.92 m/s

3. 2.38 kg m/s, 0°

4. 48 N, 180°

5. CM is 4.673 × 10⁶ m away from center of earth

12.5 m/s, 1013 m/s

6. 29793 m/s

7. earth: 29781 m/s to 29806 m/s

moon: 28780 m/s to 30806 m/s

8. 38.9 km/s at aphelion; 59.0 km/s at perihelion

9. x = 114 m; v = 30.5 m/s, 319.1°

10. 70.5 kW

11. a. U(x) = 2 x3 - 5 x

b. -0.203 m < x < 1.67 m

c. v > 2.46 m/s