A 1.25-kg wooden block rests on a table over a large hole as in the figure. A 5.00-g bullet with an initial velocity v_{i} is fired upward into the bottom of the block and remains in the block after the collision. The block and bullet rise to a maximum height of 22.0 cm. Calculate the initial velocity of the bullet from the information provided.

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A cannon is rigidly attached to a carriage, which can move along horizontal rails but is connected to a post by a large spring, initially unstretched and with force constant k = 2.00 X 10^{4} N/m, as shown in the figure. The cannon fires a 200-kg projectile at a velocity of 125 m/s directed 45.0° above the horizontal. Assuming that the mass of the cannon and its carriage is 5000 kg,

(a) Find the recoil speed of the cannon.

(b) Determine the maximum extension of the spring.

(c) Find the maximum force the spring exerts on the carriage.

(d) Consider the system consisting of the cannon, carriage, and projectile. Is the momentum of this system conserved during the firing? Why or why not?

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A bullet of mass m = 8.00 g is fired into a block of mass M = 250 g that is initially at rest at the edge of a table of height h = 1.00 m. The bullet remains in the block, and after the impact the block lands d = 2.00 m from the bottom of the table. Determine the initial speed of the bullet.

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A 3.00-kg steel ball strikes a wall with a speed of 10.0 m/s at an angle of *θ* = 60.0° with the surface. It bounces off with the same speed and angle as shown in the figure. If the ball is in contact with the wall for 0.200 s, what is the average force exerted by the wall on the ball?

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A rod of length 30.0 cm has linear density (mass per length) given by λ = 50.0 + 20.0x where x is the distance from one end, measured in meters, and λ is in grams/meter.

(a) What is the mass of the rod?

(b) How far from the x = 0 end is its center of mass?

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A uniform piece of sheet metal is shaped as shown in the figure. Compute the x and y coordinates of the center of mass of the piece.

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A 10.0-g bullet is fired into a stationary block of wood having mass m = 5.00 kg. The bullet imbeds into the block. The speed of the bullet-plus-wood combination immediately after the collision is 0.600 m/s. What was the original speed of the bullet?

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After a 0.300-kg rubber ball is dropped from a height of 1.75 m, it bounces off a concrete floor and rebounds to a height of 1.50 m. (a) Determine the magnitude and direction of the impulse delivered to the ball by the floor. (b) Estimate the time the ball is in contact with the floor and use this estimate to calculate the average force the floor exerts on the ball.

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A 65.0-kg boy and his 40.0-kg sister, both wearing roller blades, face each other at rest. The girl pushes the boy hard, sending him backward with velocity 2.90 m/s toward the west. Ignore friction.

(a) Describe the subsequent motion of the girl.

(b) How much potential energy in the girl’s body is converted into mechanical energy of the boy–girl system?

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A 3.00-kg particle has a velocity of (13.00 î + 4.00 ĵ) m/s.

(a) Find its x and y components of momentum.

(b) Find the magnitude and direction of its momentum.

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An object has a kinetic energy of 275 J and a momentum of magnitude 25.0 kg•m/s. Find the speed and mass of the object.

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An inclined plane of angle *θ* = 20.0° has a spring of force constant *k* = 500 N/m fastened securely at the bottom so that the spring is parallel to the surface as shown in Figure P7.63. A block of mass *m* = 2.50 kg is placed on the plane at a distance *d* = 0.300 m from the spring. From this position, the block is projected downward toward the spring with speed *v* = 0.750 m/s. By what distance is the spring compressed when the block momentarily comes to rest?

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For the potential energy curve shown in Figure P7.52, (a) determine whether the force *F*_{x} is positive, negative, or zero at the five points indicated. (b) Indicate points of stable, unstable, and neutral equilibrium. (c) Sketch the curve for *F*_{x} versus x from x = 0 to x = 9.5 m.

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A single conservative force acts on a 5.00-kg particle within a system due to its interaction with the rest of the system. The equation *F*_{x} = 2*x* + 4 describes the force, where *F** _{x} *is in newtons and

(a) the work done by this force on the particle

(b) the change in the potential energy of the system

(c) the kinetic energy the particle has at

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A 400-N child is in a swing that is attached to a pair of ropes 2.00 m long. Find the gravitational potential energy of the child–Earth system relative to the child’s lowest position when (a) the ropes are horizontal, (b) the ropes make a 30.0° angle with the vertical, and (c) the child is at the bottom of the circular arc.

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A 0.20-kg stone is held 1.3 m above the top edge of a water well and then dropped into it. The well has a depth of 5.0 m. Relative to the configuration with the stone at the top edge of the well, what is the gravitational potential energy of the stone–Earth system (a) before the stone is released and (b) when it reaches the bottom of the well? (c) What is the change in gravitational potential energy of the system from release to reaching the bottom of the well?

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A 4.00-kg particle is subject to a net force that varies with position as shown in the figure. The particle starts from rest at x 5 0. What is its speed at (a) x = 5.00 m, (b) x = 10.0 m, and (c) x = 15.0 m?

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A 0.600-kg particle has a speed of 2.00 m/s at point **A** and kinetic energy of 7.50 J at point **B**. What is (a) its kinetic energy at **A**, (b) its speed at **B**, and (c) the net work done on the particle by external forces as it moves from **A** to **B**?

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(a) A 3.00-kg object has a velocity (6.00 i - 2.00 j) m/s. What is its kinetic energy at this moment?

(b) What is the net work done on the object if its velocity changes to (8.00 i + 4.00 j) m/s?

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When a 4.00-kg object is hung vertically on a certain light spring that obeys Hooke’s law, the spring stretches 2.50 cm. If the 4.00-kg object is removed, (a) how far will the spring stretch if a 1.50-kg block is hung on it? (b) How much work must an external agent do to stretch the same spring 4.00 cm from its unstretched position?

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A particle is subject to a force *F*_{x} that varies with position as shown in Figure P7.15. Find the work done by the force on the particle as it moves (a) from *x* = 0 to *x* = 5.00 m, (b) from *x* = 5.00 m to *x* = 10.0 m, and (c) from *x* = 10.0 m to *x* = 15.0 m. (d) What is the total work done by the force over the distance *x* = 0 to *x* = 15.0 m?

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The force acting on a particle varies as shown in Figure P7.14. Find the work done by the force on the particle as it moves (a) from *x* = 0 to *x* = 8.00 m, (b) from *x* = 8.00 m to *x* = 10.0 m, and (c) from *x* = 0 to *x* = 10.0 m.

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A uniform chain of length 8.00 m initially lies stretched out on a horizontal table. Assuming the coefficient of static friction between chain and table is 0.600, show that the chain will begin to slide off the table if at least 3.00 m of it hangs over the edge of the table.

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A block of mass m_{1} = 20.0 kg is connected to a block of mass m_{2} = 30.0 kg by a massless string that passes over a light, frictionless pulley. The 30.0-kg block is connected to a spring that has negligible mass and a force constant of k = 250 N/m as shown in the figure. The spring is unstretched when the system is as shown in the figure, and the incline is frictionless. The 20.0-kg block is pulled a distance h = 20.0 cm down the incline of angle θ = 40.0° and released from rest. Find the speed of each block when the spring is again unstretched.

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A 10.0-kg block is released from rest at point A in Figure P8.63. The track is frictionless except for the portion between points B and C, which has a length of 6.00 m. The block travels down the track, hits a spring of force constant 2 250 N/m, and compresses the spring 0.300 m from its equilibrium position before coming to rest momentarily. Determine the coefficient of kinetic friction between the block and the rough surface between points B and C.

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