1. Problems
Ch(1-3)

2. [10] The position of a particle moving under uniform acceleration is some function of time and the acceleration. Suppose we write this position s = kam tn where , k is a dimensionless constant. Show by dimensional analysis that this expression is satisfied if if m = 1 and n = 2
Solution
and

3. [17] Newton’s law of universal gravitation is represented by
Here F is the magnitude of the gravitational force exerted by one small object on another , M and m are the masses of the objects, and r is a distance. Force has the SI units kg ·m/ s2. What are the SI units of the proportionality constant G?
Solution:

4. [23] A solid piece of lead has a mass of 23. 94 g and a volume of 2
[23] A solid piece of lead has a mass of g and a volume of 2.10 cm3. From these data, calculate the density of lead in SI units (kg/ m3).
Solution
One centimeter (cm) equals 0.01 m.
One kilometer (km) equals 1000 m.
One inch equals 2.54 cm
One foot equals 30 cm…
Example:

5. [2] If the rectangular coordinates of a point are given by (2, y) and its polar coordinates are ( r , 30°), determine y and r.
Solution:
then
then
[4] Two points in the xy plane have Cartesian coordinates (2.00, -4.00) m and ( -3.00, 3.00) m. Determine (a) the distance between these points and (b) their polar coordinates. a)
Solution:

6. For (2,-4) the polar coordinate is (2,2√5) sinceb
For (2,-4) the polar coordinate is (2,2√5) since

7. [12] Vector A has a magnitude of 8. 00 units and makes an angle of 45
[12] Vector A has a magnitude of 8.00 units and makes an angle of 45.0 ° with the positive x axis. Vector B also has a magnitude of 8.00 units and is directed along the negative x axis. Using graphical methods, find (a) the vector sum A + B and (b) the vector difference A - B.
Solution:

8. [25] Given the vectors A = 2. 00 i +6. 00 j and B = 3. 00 i - 2
[25] Given the vectors A = 2.00 i j and B = 3.00 i j, (a) draw the vector sum , C = A + B and the vector difference D = A - B. (b) Calculate C and D, first in terms of unit vectors and then in terms of polar coordinates, with angles measured with respect to the , +x axis.
Solution:

9. [29] Consider the two vectors A = 3 i - 2 j and B = i - 4 j
[29] Consider the two vectors A = 3 i - 2 j and B = i - 4 j. Calculate (a) A + B, (b) A - B, (c) │A + B│, (d) │A - B│, and (e) the directions of A + B and A - B.
Solution:
Direction of A+B
Direction of A-B

10. [41] The vector A has x, y, and , z components of 8. 00, 12. 0, and -4
[41] The vector A has x, y, and , z components of 8.00, 12.0, and units, respectively. (a) Write a vector expression for A in unit vector notation. (b) Obtain a unit vector expression for a vector B four time the length of A pointing in the same direction as A. (c) Obtain a unit vector expression for a vector C three times the length of A pointing in the direction opposite the direction of A.
Solution: a) b) c)

11. [*] Three displacement vectors of a croquet ball are shown in Figure, where A = 20.0 units, B = 40.0 units, and C = 30.0 units. Find the magnitude and direction of the resultant displacement
Solution:

12. Example:
Find the magnitude and the direction of resultant force

13. [3] The position versus time for a certain particle moving along the x axis is shown in Figure. Find the average velocity in the time intervals
(a) 0 to 2 s, (b) 2 to 4 s, (c) 4 s to 5 s, (d) 4 s to 7 s, (e) 0 to 8 s.
a) 0 to 2 s
b) 2 to 4 s
c) 4 to 5 s
d) 4 to 7 s

14. [*] A position-time graph for a particle moving along the x axis is shown in Figure (a) Find the average velocity in the time interval t = 1.50 s to t = 4 s. (b) At what value of t is the velocity zero?
Solution:
(a)
(b)
At time t = 4 sec

15. [17] A particle moves along the x axis
[17] A particle moves along the x axis. Its position is given by the equation x= 2 t + 3t 2 with x in meters and t in seconds. Calculate the velocity and acceleration at t=3s.
Solution:
First, the instantaneous velocity at t=3s
The instantaneous acceleration at t=3s

16. [*] Consider the motion of the object whose velocity-time graph is given in the diagram
1- What is the acceleration of the object between times t=0 to 2s?
2- What is the acceleration of the object between times t=10 to 12s?
3- What is the net displacement of the object between times 0 to 16s?
Solution: 1- 2- 3-
equals the area under the v-t graph =100m2 how? the net displacement  

17. [21] A particle moves along the x axis according to the equation x = 2+ 3t - t 2 where , x is in meters and t is in seconds. At t= 3 s, find (a) the position of the particle, (b) its velocity, and (c) its acceleration.
Solution:
(a) the position of the particle
(b) its velocity,
(c) its acceleration

18. Let the origin be at the initial position x0=0
[*] A truck covers 40m in 8.5 s while smoothly slowing down to a final speed of m/s. (a) Find its original speed. (b) Find its acceleration.
Solution:
Let the origin be at the initial position x0=0
(a)
(b)

19. Example:  A boy throws a ball upward from the top of a building with an initial velocity of 20.0 m/s. The building is 50 meters high. Determine (a) the time needed for the ball to reach its maximum height (b) the maximum height (c) the time needed for the ball to reach the level of the thrower (d) the velocity of the stone at this instant (e) the velocity and position of the ball after 5.00 s (f) the velocity of the ball when it reaches the bottom of the building
Solution:
Given Information:
g = 9.8 m/s2
vo = 20 m/s
yo= 50 m
(a)  At maximum height v  = 0, so using

20. (b) the maximum height
(c) the time needed for the ball to reach the level of the thrower
Note :the time to reach the top and back down to the thrower's level is the same so 2(2.04) = 4.08 seconds
(d) the velocity of the ball at this instant
Which is the same speed of the original throw upward, except now the negative implies that the ball is moving with a speed of 20 m/s downward.

21. (e) the velocity and position of the ball after 5.00 s
(f)  The velocity of the ball just at the moment it reaches the ground,
But, the context of the problem would have us choose, m/s because the ball is moving downward just as it hits the ground.

22. Example:  Suppose a spacecraft is traveling with a speed of 3250 m/s, and it slows down by firing its retro rockets, so that a= -10 m/s2.  What is the velocity of the spacecraft after it has traveled 215km? 
Solution:

23. [1] A car moved 20 km East and 60 km West in 2 hours
[1] A car moved 20 km East and 60 km West in 2 hours. What is its average velocity? [ 20m/s ] [2] A car moved 20 km East and 70 km West. What is the distance? [ 90m/s ]
[3] If a car accelerates from 5 m/s to 15 m/s in 2 seconds, what is the car's average acceleration? Answer [ 5m/s2 ]
[4] How long does it take to accelerate an object from rest to 10 m/s if the acceleration was 2 m/s? Answer [ 5s ]
[5] If a car accelerated from 5 m/s to 25 m/s in 10 seconds, how far will it travel? Answer [ 150m ]
[6] What is the displacement of a car whose initial velocity is 5 m/s and then accelerated 2 m/s2 for 10 seconds? Answer [ 150m ]
[7] What is the final velocity of a car that accelerated 10 m/s2 from rest and traveled 180m? Answer [ 60m/s ]
[8] How long will it take for an apple falling from a 29.4m-tall tree to hit the ground? Answer [ 2.4s ]

24. Problems
Chapter 4,5

25. Problem 1
A ball from a tower of height 30 m is projected down at an angle of 30° from the horizontal with a speed of 10 m/s. How long does it take the ball to reach the ground. (Consider g = 10 m/s2 ) :
Solution:
By using the equation
Neglecting negative value of time, t = -2 s

26. Problem 2
On a hot summer day a young girl swings on a rope above the local swimming hole. When she lets go of the rope her initial velocity is 2.25 m/s at an angle of 35.0° above the horizontal. If she is in flight for 1.60 s, how high above the water was she when she let go of the rope?
Solution:
v0 = 2.25 m/s
v0x = v0 cos(35) = 1.8 m/s
v0y = v0 sin(35) = 1.3 m/s

27. By using
y = y0 +v0y t – (1/2) g t2
y-y0 = m
y =0
y0 = m

28. (1) What is the time t for the projectile to reach the ground?
Problem 3
A projectile is launched horizontally with an initial speed v0 from a height y0. Neglect aerodynamic friction.
v0 = 3.5 m/s y0 = 4.0 m
Solution:
(1) What is the time t for the projectile to reach the ground? v0 x y y0

29. (2) What is the horizontal component of the projectile’s velocity when it hits the ground?
(3) What is the speed of the projectile when it hits the ground?

30. Problem 4
Suppose we have a situation as we did in a lab experiment, in which a hanging mass and cart are connected by a string over a pulley.
The hanging mass is 0.20 kg, but we don't know the mass of the cart. After the cart is released and the objects are moving, the tension in the string is found to be 1.80 N.

31. (A) Find the acceleration of the hanging mass
(A) Find the acceleration of the hanging mass. First, draw a free-body diagram

32. (B) Find the mass of the cart, assuming that there is no friction.

33. (C) find the mass of the cart assuming that there is a 0
(C) find the mass of the cart assuming that there is a 0.50 N frictional force acting on the cart.

34. Problem 5
Find the tensions in each string in the figure at right. They are attached to a wall and ceiling, as shown, and are holding up a mass of 10 kg
Solution:
Knowing T1, we can find T2:

35. Problem 6
We show below two blocks connected by a light string and pulled by a string in front. The “front” block has a mass of 5.0 kg and the other block’s mass is 2.0 kg. The large block slides on frictionless surface. The back, smaller block has a rough bottom surface which creates a frictional force of 6.0 N. The tension T1 in the front string is 20.0 N. Determine the tension T2 in the middle string
Solution:

36. Problem 7
A 20-kg box is sitting on the floor and someone is pulling on it with a rope. The tension in the rope is 60 N at an angle 60◦ above the horizontal. However, the box does not move. It just sits there, held by static friction.
(a) Draw a free-body diagram of the box.

37. (b) Find the normal force that the floor exerts on the box.

38. Problem 8
Two forces F1 and F2 act on a 5.00-kg object. If F1 = 20.0 N and F2 = 15.0 N, find the accelerations in (a) and (b) of Figure
Problem 9
A 3.00-kg object is moving in a plane, with its x and y coordinates given by x = 5t2 + 1 and y =3t 3 + 2, where x and y are in meters and t is in seconds. Find the magnitude of the net force acting on this object at t = 2.00 s

39. Quiz
(1) A projectile is fired in such a way that its horizontal range is equal to three times its maximum height. What is the angle of projection?
(2) An inclined plane makes an angle 20◦ with the horizontal. A 0.20-kg bar of soap slides down the incline, and experiences a frictional force of 0.20 N. (A) Draw a free-body diagram of the bar of soap. (B) Find the net force on the bar of soap. (C) Calculate the rate of acceleration of the bar of soap