# AP Physics C: Mechanics Practice (Systems of Particles and Linear Momentum)

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1 AP Physics C: Mechanics Practice (Systems of Particles and Linear Momentum) 1980M2. A block of mass m slides at velocity v o across a horizontal frictionless surface toward a large curved movable ramp of mass 3m as shown in Figure 1. The ramp, initially at rest, also can move without friction and has a smooth circular frictionless face up which the block can easily slide. When the block slides up the ramp, it momentarily reaches a maximum height as shown in Figure II and then slides back down the frictionless face to the horizontal surface as shown in Figure III. a. Find the velocity v 1 of the moving ramp at the instant the block reaches its maximum height. b. To what maximum height h does the center of mass of the block rise above its original height? c. Determine the final speed v f of the ramp and the final speed v' of the block after the block returns to the level surface. State whether the block is moving to the right or to the left.

2 1985M1. A projectile is launched from the top of a cliff above level ground. At launch the projectile is 35 meters above the base of the cliff and has a velocity of 50 meters per second at an angle 37 with the horizontal. Air resistance is negligible. Consider the following two cases and use g = 10 m/s 2, sin 37 = 0.60, and cos 37 = Case I: The projectile follows the path shown by the curved line in the following diagram. a. Calculate the total time from launch until the projectile hits the ground at point C. b. Calculate the horizontal distance R that the projectile travels before it hits the ground. c. Calculate the speed of the projectile at points A, B and C. Case II: A small internal charge explodes at point B in the above diagram, causing the projectile to separate into two parts of masses 6 kilograms and 10 kilograms. The explosive force on each part is horizontal and in the plane of the trajectory. The 6-kilogram mass strikes the ground at point D, located 30 meters beyond point C, where the projectile would have landed had it not exploded The 10-kilogram mass strikes the ground at point E. Find x.

3 1991M1. A small block of mass 2m initially rests on a track at the bottom of the circular, vertical loop-the-loop shown above, which has a radius r. The surface contact between the block and the loop is frictionless. A bullet of mass m strikes the block horizontally with initial speed v o and remains embedded in the block as the block and bullet circle the loop. Determine each of the following in terms of m, v 0 r, and g. a. The speed of the block and bullet immediately after impact b. The kinetic energy of the block and bullet when they reach point P on the loop c. The minimum initial speed v min of the bullet if the block and bullet are to successfully execute a complete circuit of the loop

4 1991M3. The two blocks I and II shown above have masses m and 2m respectively. Block II has an ideal massless spring attached to one side. When block I is placed on the spring as shown. the spring is compressed a distance D at equilibrium. Express your answer to all parts of the question in terms of the given quantities and physical constants. a. Determine the spring constant of the spring Later the two blocks are on a frictionless, horizontal surface. Block II is stationary and block I approaches with a speed v o, as shown above. b. The spring compression is a maximum when the blocks have the same velocity. Briefly explain why this is so. c. Determine the maximum compression of the spring during the collision. d. Determine the velocity of block II after the collision when block I has again separated from the spring.

5 1992M1. A ball of mass 9m is dropped from rest from a height H = 5.0 meters above the ground, as shown above on the left. It undergoes a perfectly elastic collision with the ground and rebounds. At the instant that the ball rebounds, a small blob of clay of mass m is released from rest from the original height H, directly above the ball, as shown above on the right. The clay blob, which is descending, eventually collides with the ball, which is ascending. Assume that g = 10 m/s 2, that air resistance is negligible, and that the collision process takes negligible time. a. Determine the speed of the ball immediately before it hits the ground. b. Determine the time after the release of the clay blob at which the collision takes place. c. Determine the height above the ground at which the collision takes place. d. Determine the speeds of the ball and the clay blob immediately before the collision. e. If the ball and the clay blob stick together on impact, what is the magnitude and direction of their velocity immediately after the collision?

6 1993M1. A massless spring with force constant k = 400 newtons per meter is fastened at its left end to a vertical wall, as shown in Figure 1. Initially, block C (mass m c = 4.0 kilograms) and block D (mass m D = 2.0 kilograms) 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 11. (Use g = 10 m/s 2.) 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 µ = 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 blocks C and D just after they collide d. The horizontal distance the blocks move before coming to rest

7 1994M1. A 2- kilogram block and an 8- kilogram block are both attached to an ideal spring ( for which k = 200 N/m) and both are initially at rest on a horizontal frictionless surface, as shown in the diagram above. In an initial experiment, a 100- gram (0.1 kg) ball of clay is thrown at the 2- kilogram block. The clay is moving horizontally with speed v when it hits and sticks to the block. The 8- kilogram block is held still by a removable stop. As a result, the spring compresses a maximum distance of 0.4 meters. a. Calculate the energy stored in the spring at maximum compression. b. Calculate the speed of the clay ball and 2- kilogram block immediately after the clay sticks to the block but before the spring compresses significantly. c. Calculate the initial speed v of the clay. In a second experiment, an identical ball of clay is thrown at another identical 2- kilogram block, but this time the stop is removed so that the 8- kilogram block is free to move. d. State whether the maximum compression of the spring will be greater than, equal to, or less than 0.4 meter. Explain briefly. e. State the principle or principles that can be used to calculate the velocity of the 8- kilogram block at the instant that the spring regains its original length. Write the appropriate equation(s) and show the numerical substitutions, but do not solve for the velocity.

8 1995M1. A 5-kilogram ball initially rests at the edge of a 2-meter-long, 1.2-meter-high frictionless table, as shown above. A hard plastic cube of mass 0.5 kilogram slides across the table at a speed of 26 meters per second and strikes the ball, causing the ball to leave the table in the direction in which the cube was moving. The figure below shows a graph of the force exerted on the ball by the cube as a function of time. a. Determine the total impulse given to the ball. b. Determine the horizontal velocity of the ball immediately after the collision. c. Determine the following for the cube immediately after the collision. i. Its speed ii. Its direction of travel (right or left), if moving d. Determine the kinetic energy dissipated in the collision. e. Determine the distance between the two points of impact of the objects with the floor.

9 1997M2. An open- top railroad car (initially empty and of mass M o) rolls with negligible friction along a straight horizontal track and passes under the spout of a sand conveyor. When the car is under the conveyor, sand is dispensed from the conveyor in a narrow stream at a steady rate ΔM/Δt = C and falls vertically from an average height h above the floor of the railroad car. The car has initial speed v o and sand is filling it from time t = 0 to t = T. Express your answers to the following in terms of the given quantities and g. a. Determine the mass M of the car plus the sand that it catches as a function of time t for C < t < T. b. Determine the speed v of the car as a function of time t for 0 < t < T. c. i. Determine the initial kinetic energy K i of the empty car. ii. Determine the final kinetic energy K f of the car and its load. iii. Is kinetic energy conserved? Explain why or why not. d. Determine expressions for the normal force exerted on the car by the tracks at the following times. i. Before t = 0 ii. For 0 < t < T iii. After t = T

10 1998M1. Two gliders move freely on an air track with negligible friction, as shown above. Glider A has a mass of 0.90 kg and glider B has a mass of 0.60 kg. Initially, glider A moves toward glider B, which is at rest. A spring of negligible mass is attached to the right side of glider A. Strobe photography is used to record successive positions of glider A at 0.10 s intervals over a total time of 2.00 s, during which time it collides with glider B. The following diagram represents the data for the motion of glider A. Positions of glider A at the end of each 0.10s interval are indicated by the symbol A against a metric ruler. The total elapsed time t after each 0.50 s is also indicated. a. Determine the average speed of glider A for the following time intervals. i s to 0.30 s ii s to 1.10 s iii s to 1.90 s b. On the axes below, sketch a graph, consistent with the data above, of the speed of glider A as a function of time t for the 2.00 s interval.

11 c. i. Use the data to calculate the speed of glider B immediately after it separates from the spring. ii. On the axes below, sketch a graph of the speed of glider B as a function of time t. A graph of the total kinetic energy K for the two- glider system over the 2.00 s interval has the following shape. K o is the total kinetic energy of the system at time t = 0. d. i. Is the collision elastic? Justify your answer. ii. Briefly explain why there is a minimum in the kinetic energy curve at t = 1.00 s.

12 1999M1 In a laboratory experiment, you wish to determine the initial speed of a dart just after it leaves a dart gun. The dart, of mass m, is fired with the gun very close to a wooden block of mass M0 which hangs from a cord of length l and negligible mass, as shown above. Assume the size of the block is negligible compared to l, and the dart is moving horizontally when it hits the left side of the block at its center and becomes embedded in it. The block swings up to a maximum angle from the vertical. Express your answers to the following in terms of m, M0, l, θmax, and g. a. Determine the speed v0 of the dart immediately before it strikes the block. b. The dart and block subsequently swing as a pendulum. Determine the tension in the cord when it returns to the lowest point of the swing. c. At your lab table you have only the following additional equipment. Meter stick Stopwatch Set of known masses Protractor 5 m of string Five more blocks of mass M0 Spring Without destroying or disassembling any of this equipment, design another practical method for determining the speed of the dart just after it leaves the gun. Indicate the measurements you would take, and how the speed could be determined from these measurements. d. The dart is now shot into a block of wood that is fixed in place. The block exerts a force F on the dart that is proportional to the dart's velocity v and in the opposite direction, that is F = - bv, where b is a constant. Derive an expression for the distance L that the dart penetrates into the block, in terms of m, v0, and b.

13 2001M1. A motion sensor and a force sensor record the motion of a cart along a track, as shown above. The cart is given a push so that it moves toward the force sensor and then collides with it. The two sensors record the values shown in the following graphs. a. Determine the cart's average acceleration between t = 0.33 s and t = 0.37 s. b. Determine the magnitude of the change in the cart's momentum during the collision. c. Determine the mass of the cart. d. Determine the energy lost in the collision between the force sensor and the cart

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