Conservation of Energy ⇒ KEi + PEi = KEf + PEf + W
Set h = 0 at the height of the billiard balls. ⇒ All PE = 0. W = 0 (elastic).
⇒ ½m1v1i² + ½m2v2i² = ½m1v1f² + ½m2v2f² ⇒ m1v1i² + m2v2i² = m1v1f² + m2v2f²
⇒ (0.5 kg)(0.15 m/sec)² + (0.35 kg)(-0.2 m/sec)² = 0.02525 J = m1v1f² + m2v2f²
⇒ 0.02525 = (0.5)v1f² + (0.35)v2f² (From Cons. of Energy, two unknowns)
Conservation of Momentum ⇒ m1v1i + m2v2i = m1v1f + m2v2f
⇒ (0.5 kg)(0.15 m/sec) - (0.35 kg)(0.2 m/sec) = 0.005 kg-m/sec = m1v1f + m2v2f
⇒ 0.005 = (0.5)v1f + (0.35)v2f (From Cons. of Momentum, two unknowns)
Substitute v2f = (0.005 - (0.5)v1f)/(0.35) into Cons. of Energy Eqn.
⇒ 0.02525 = (0.5)v1f² + (0.35)[(0.005 - (0.5)v1f)/(0.35)]²
⇒ (1.2143)v1f² - (0.01429)v1f - 0.02518 = 0 ⇒ v1f = -0.15 m/sec or 0.138 m/sec
⇒ v1f = -0.15 m/sec and v2f = 0.229 m/sec
2. A 10.5 g bullet strikes a pendulum that consists of a block of wood of mass 3.00 kg suspended by a cord. The bullet gets embedded in the block. How fast was the bullet traveling just before impact to raise the block by 0.220 m?
3. Sheila, running 5.3m/s, grabs a vine hanging vertically from a tall tree.
a. How high can she swing upward?
b. Does the length of vine affect the answer?
4. A roller coaster at the top of a 39.0 m high vertical loop is traveling 13.8 m/s. Find the maximum speed of the cars as they move through the bottom of the loop.
5. Analyze the motion of a simple swinging pendulum in terms of energy, (a) ignoring friction; and (b) taking friction into account. Explain why a grandfather clock has to be wound up.
6. A ball is attached to a horizontal cord of length L whose other end is fixed.
a. If the ball is released, what will be its speed at the lowest point of its path?
b. A peg is located a distance h directly below the point of attachment of the cord. If h= 0.080L, what will be the speed of the ball when it reaches the top of the circular path about the peg?
7. A projectile is fired at an upward angle of 45.0 degree from the top of a 265 m cliff with a speed of 185 m/s. What will be its maximum speed of impact with the ground below?
8. If a projectile is launched from Earth with a speed equal to the escape speed, how high above the Earth's surface is it when its speed is one third the escape speed?
9. Two pieces of space debris, each with a mass of 0.116 kg, are separated by a distance of 380 m. If re released from rest, what speed do they have when their separation has decreased to 171 m? Ignore the gravitational effects from any other objects.
10. A baseball is thrown first with an initial upward velocity of + 4.0 m/s. Later, it is thrown from the same height but with and initial downward velocity of -3.0m/s. How do the impact velocities of the baseball with the ground differ? What is its acceleration in each case?
11. A 200 g ball is thrown upwards with an initial kinetic energy of 10 Joules. What maximum height will the ball attain?
12. Bruce grasps the end of a 20.0 m long rope attached to a tree and swings. If the rope starts at an angle 35 degrees with the vertical, what is Bruce's speed at the bottom of the swing?
13. Jack and Jill, whose total mass is 120 kg, sit on a swing at the end of a 5m long rope. Initially the rope attached to their swing makes an angle of 36 degrees with the horizontal. At the bottom of the arc, Jill, whose mass is 52 kg, steps off. What is the maximum height Jack can reach as the swing continues?
14. A 1.9-kg block slides down a curved, frictionless ramp. The top of the ramp is 1.5 m above ground; the bottom of the ramp is 0.25 m above the ground. The block leaves the bottom of the ramp moving horizontally.
a. What horizontal distance away from the base of the ramp does it land?
b. Suppose friction on the ramp does -9.7 J of work on the block. What horizontal distance away from the base of the ramp does it land?
15. A ball bounces upward from the ground with a speed of 16 m/s and hits a wall with a speed of 12 m/s How high above the ground does the ball hit the wall? Ignore air resistance.
16. From what height would a car have to be dropped to have the same kinetic energy that it has when being driven at 100 km/h?
17. A 135 m long ramp is to be built for a ski jump. If a skier starting from rest at the top is to have a speed no faster than 19m/s at the bottom, what should be the maximum angle of inclination?
18. Show that the escape speed from the surface of a planet of uniform density is directly proportional to the radius of the planet.
19. Two objects, m1 = 4.50kg and m2 = 3.00kg, are connected by a light string passing over a light frictionless pulley. The object of the mass 4.50kg is released from rest 4.50m above the ground. Using the principle of conservation of energy, determine the speed of the 3.00kg object just as the 4.50kg object hits the ground.
(a) What is the escape speed on a spherical asteroid whose radius is 525km and whose gravitational acceleration at the surface is 2.7m/s2?
(b.)How far from the surface will a particle go if it leaves the asteroid's surface with a vertical speed of 1000m/s?
21. In a looping the loop setup, an object of mass m is released from rest from A with the initial height h and loops the loop in the circular track of radius R.
a) Write an expression for the initial mechanical energy at A in terms of m, g, h
b) Write an expression for energy at point B at the top of the vertical circle in terms of mass m, velocity v, radius R and g
c) If in the absence of friction the object just manages to loop the loop without losing contact with the track, what is the minimum height h from which you will need to release the object? Write the expression in terms of R .
22. A 75 kg parcel falls out of a window to a sidewalk 1 m below.
a. With what speed does it impact the pavement?
b. If the packaging provides 0.50 cm of cushioning, calculate the average force exerted on the parcel by the ground in this situation.
23. Roy was transporting balls in the trunk of a car to a clubhouse. Two boxes on the floor of the trunk each contained an equal number of balls. The balls were identical except that all the balls in one box were dimpled, while all the balls in the other box were smooth. Upon arriving, Roy realized there were no lids on the boxes, and found balls all over the trunk of the car. He observed that more dimpled balls escaped than smooth balls. Why would more dimpled balls escape than smooth balls? There was nothing else in the trunk.TOP OF PAGE