1. 13
VECTOR FUNCTIONS

2. Motion in Space: Velocity and Acceleration
VECTOR FUNCTIONS
13.4
Motion in Space: Velocity and Acceleration
In this section, we will learn about:
The motion of an object
using tangent and normal vectors.

3. MOTION IN SPACE: VELOCITY AND ACCELERATION
Here, we show how the ideas of tangent and normal vectors and curvature can be used in physics to study:
The motion of an object, including its velocity and acceleration, along a space curve.

4. VELOCITY AND ACCELERATION
In particular, we follow in the footsteps of Newton by using these methods to derive Kepler’s First Law of planetary motion.

5. VELOCITY
Suppose a particle moves through space so that its position vector at time t is r(t).

6. VELOCITY
Vector 1
Notice from the figure that, for small values of h, the vector approximates the direction of the particle moving along the curve r(t).

7. VELOCITY
Its magnitude measures the size of the displacement vector per unit time.
IS THE PREVIOUS IMAGE NEEDED HERE TOO?

8. VELOCITY
The vector 1 gives the average velocity over a time interval of length h.

9. Its limit is the velocity vector v(t) at time t :
Equation 2
Its limit is the velocity vector v(t) at time t :

10. VELOCITY VECTOR
Thus, the velocity vector is also the tangent vector and points in the direction of the tangent line.

11. SPEED
The speed of the particle at time t is the magnitude of the velocity vector, that is, |v(t)|.

12. = rate of change of distance with respect to time
SPEED
This is appropriate because, from Equation 2 and from Equation 7 in Section 13.3, we have:
= rate of change of distance with respect to time

13. ACCELERATION
As in the case of one-dimensional motion, the acceleration of the particle is defined as the derivative of the velocity: a(t) = v’(t) = r”(t)

14. VELOCITY & ACCELERATION
Example 1
The position vector of an object moving in a plane is given by: r(t) = t3 i + t2 j
Find its velocity, speed, and acceleration when t = 1 and illustrate geometrically.

15. VELOCITY & ACCELERATION
Example 1
The velocity and acceleration at time t are:
v(t) = r’(t) = 3t2 i + 2t j
a(t) = r”(t) = 6t I + 2 j

16. VELOCITY & ACCELERATION
Example 1
The speed at t is:

17. VELOCITY & ACCELERATION
Example 1
When t = 1, we have: v(1) = 3 i + 2 j a(1) = 6 i + 2 j |v(1)| =

18. VELOCITY & ACCELERATION
Example 1
These velocity and acceleration vectors are shown here.

19. VELOCITY & ACCELERATION
Example 2
Find the velocity, acceleration, and speed of a particle with position vector r(t) = ‹t2, et, tet›

20. VELOCITY & ACCELERATION
Example 2

21. VELOCITY & ACCELERATION
The figure shows the path of the particle in Example 2 with the velocity and acceleration vectors when t = 1.

22. VELOCITY & ACCELERATION
The vector integrals that were introduced in Section 13.2 can be used to find position vectors when velocity or acceleration vectors are known, as in the next example.

23. VELOCITY & ACCELERATION
Example 3
A moving particle starts at an initial position r(0) = ‹1, 0, 0› with initial velocity v(0) = i – j + k
Its acceleration is a(t) = 4t i + 6t j + k
Find its velocity and position at time t.

24. VELOCITY & ACCELERATION
Example 3
Since a(t) = v’(t), we have:
v(t) = ∫ a(t) dt = ∫ (4t i + 6t j + k) dt
=2t2 i + 3t2 j + t k + C

25. VELOCITY & ACCELERATION
Example 3
To determine the value of the constant vector C, we use the fact that v(0) = i – j + k
The preceding equation gives v(0) = C.
So, C = i – j + k

26. VELOCITY & ACCELERATION
Example 3
It follows:
v(t) = 2t2 i + 3t2 j + t k + i – j + k
= (2t2 + 1) i + (3t2 – 1) j + (t + 1) k

27. VELOCITY & ACCELERATION
Example 3
Since v(t) = r’(t), we have:
r(t) = ∫ v(t) dt
= ∫ [(2t2 + 1) i + (3t2 – 1) j + (t + 1) k] dt
= (⅔t3 + t) i + (t3 – t) j + (½t2 + t) k + D

28. VELOCITY & ACCELERATION
Example 3
Putting t = 0, we find that D = r(0) = i.
So, the position at time t is given by:
r(t) = (⅔t3 + t + 1) i + (t3 – t) j + (½t2 + t) k

29. VELOCITY & ACCELERATION
The expression for r(t) that we obtained in Example 3 was used to plot the path of the particle here for 0 ≤ t ≤ 3.

30. VELOCITY & ACCELERATION
In general, vector integrals allow us to recover:
Velocity, when acceleration is known
Position, when velocity is known

31. VELOCITY & ACCELERATION
If the force that acts on a particle is known, then the acceleration can be found from Newton’s Second Law of Motion.

32. VELOCITY & ACCELERATION
The vector version of this law states that if, at any time t, a force F(t) acts on an object of mass m producing an acceleration a(t), then
F(t) = ma(t)

33. VELOCITY & ACCELERATION
Example 4
An object with mass m that moves in a circular path with constant angular speed ω has position vector r(t) = a cos ωt i + a sin ωt j
Find the force acting on the object and show that it is directed toward the origin.

34. VELOCITY & ACCELERATION
Example 4
To find the force, we first need to know the acceleration:
v(t) = r’(t) = –aω sin ωt i + aω cos ωt j
a(t) = v’(t) = –aω2 cos ωt i – aω2 sin ωt j

35. VELOCITY & ACCELERATION
Example 4
Therefore, Newton’s Second Law gives the force as:
F(t) = ma(t)
= –mω2 (a cos ωt i + a sin ωt j)

36. VELOCITY & ACCELERATION
Example 4
Notice that: F(t) = –mω2r(t)
This shows that the force acts in the direction opposite to the radius vector r(t).

37. VELOCITY & ACCELERATION
Example 4
Therefore, it points toward the origin.

38. Such a force is called a centripetal (center-seeking) force.
CENTRIPETAL FORCE
Example 4
Such a force is called a centripetal (center-seeking) force.

39. VELOCITY & ACCELERATION
Example 5
A projectile is fired with:
Angle of elevation α
Initial velocity v0

40. VELOCITY & ACCELERATION
Example 5
Assuming that air resistance is negligible and the only external force is due to gravity, find the position function r(t) of the projectile.

41. VELOCITY & ACCELERATION
Example 5
What value of α maximizes the range (the horizontal distance traveled)?

42. VELOCITY & ACCELERATION
Example 5
We set up the axes so that the projectile starts at the origin.

43. VELOCITY & ACCELERATION
Example 5
As the force due to gravity acts downward, we have: F = ma = –mg j where g = |a| ≈ 9.8 m/s2.
Therefore, a = –g j

44. VELOCITY & ACCELERATION
Example 5
Since v(t) = a, we have:
v(t) = –gt j + C
where C = v(0) = v0.
Therefore, r’(t) = v(t) = –gt j + v0

45. VELOCITY & ACCELERATION
Example 5
Integrating again, we obtain:
r(t) = –½ gt2 j + t v0 + D
However, D = r(0) = 0

46. VELOCITY & ACCELERATION
E. g. 5—Equation 3
So, the position vector of the projectile is given by: r(t) = –½gt2 j + t v0

47. VELOCITY & ACCELERATION
Example 5
If we write |v0| = v0 (the initial speed of the projectile), then
v0 = v0 cos α i + v0 sin α j
Equation 3 becomes: r(t) = (v0 cos α)t i + [(v0 sin α)t – ½gt2] j

48. VELOCITY & ACCELERATION
E. g. 5—Equations 4
Therefore, the parametric equations of the trajectory are: x = (v0 cos α)t y = (v0 sin α)t – ½gt2

49. VELOCITY & ACCELERATION
Example 5
If you eliminate t from Equations 4, you will see that y is a quadratic function of x.

50. VELOCITY & ACCELERATION
Example 5
So, the path of the projectile is part of a parabola.

51. VELOCITY & ACCELERATION
Example 5
The horizontal distance d is the value of x when y = 0.
Setting y = 0, we obtain: t = 0 or t = (2v0 sin α)/g

52. VELOCITY & ACCELERATION
Example 5
That second value of t then gives:
Clearly, d has its maximum value when sin 2α = 1, that is, α = π/4.

53. VELOCITY & ACCELERATION
Example 6
A projectile is fired with muzzle speed 150 m/s and angle of elevation 45° from a position 10 m above ground level.
Where does the projectile hit the ground?
With what speed does it do so?

54. VELOCITY & ACCELERATION
Example 6
If we place the origin at ground level, the initial position of the projectile is (0, 10).
So, we need to adjust Equations 4 by adding 10 to the expression for y.

55. VELOCITY & ACCELERATION
Example 6
With v0 = 150 m/s, α = 45°, and g = 9.8 m/s2, we have:

56. VELOCITY & ACCELERATION
Example 6
Impact occurs when y = 0, that is, t2 – t – 10 = 0
Solving this quadratic equation (and using only the positive value of t), we get:

57. VELOCITY & ACCELERATION
Example 6
Then, x ≈ (21.74) ≈ 2306
So, the projectile hits the ground about 2,306 m away.

58. VELOCITY & ACCELERATION
Example 6
The velocity of the projectile is:

59. VELOCITY & ACCELERATION
Example 6
So, its speed at impact is:

60. ACCELERATION—COMPONENTS
When we study the motion of a particle, it is often useful to resolve the acceleration into two components:
Tangential (in the direction of the tangent)
Normal (in the direction of the normal)

61. ACCELERATION—COMPONENTS
If we write v = |v| for the speed of the particle, then
Thus, v = vT

62. ACCELERATION—COMPONENTS
Equation 5
If we differentiate both sides of that equation with respect to t, we get:

63. ACCELERATION—COMPONENTS
Equation 6
If we use the expression for the curvature given by Equation 9 in Section 13.3, we have:

64. ACCELERATION—COMPONENTS
The unit normal vector was defined in Section 13.4 as N = T’/ |T’|
So, Equation 6 gives:

65. ACCELERATION—COMPONENTS
Formula/Equation 7
Then, Equation 5 becomes:

66. ACCELERATION—COMPONENTS
Equations 8
Writing aT and aN for the tangential and normal components of acceleration, we have
a = aTT + aNN
where
aT = v’ and aN = Kv2
PUT THE CORRECT FONT FOR ‘K.’

67. ACCELERATION—COMPONENTS
This resolution is illustrated here.

68. ACCELERATION—COMPONENTS
Let’s look at what Formula 7 says.

69. ACCELERATION—COMPONENTS
The first thing to notice is that the binormal vector B is absent.
No matter how an object moves through space, its acceleration always lies in the plane of T and N (the osculating plane).
Recall that T gives the direction of motion and N points in the direction the curve is turning.

70. ACCELERATION—COMPONENTS
Next, we notice that:
The tangential component of acceleration is v’, the rate of change of speed.
The normal component of acceleration is ĸv2, the curvature times the square of the speed.
PUT THE CORRECT FONT FOR ‘K.’

71. ACCELERATION—COMPONENTS
This makes sense if we think of a passenger in a car.
A sharp turn in a road means a large value of the curvature ĸ.
So, the component of the acceleration perpendicular to the motion is large and the passenger is thrown against a car door.
PUT THE CORRECT FONT FOR ‘K.’

72. ACCELERATION—COMPONENTS
High speed around the turn has the same effect.
In fact, if you double your speed, aN is increased by a factor of 4.

73. ACCELERATION—COMPONENTS
We have expressions for the tangential and normal components of acceleration in Equations 8.
However, it’s desirable to have expressions that depend only on r, r’, and r”.

74. ACCELERATION—COMPONENTS
Thus, we take the dot product of v = vT with a as given by Equation 7: v · a = vT · (v’ T + ĸv2N) = vv’ T · T + ĸv3T · N = vv’ (Since T · T = 1 and T · N = 0)
PUT THE CORRECT FONT FOR ‘K.’

75. ACCELERATION—COMPONENTS
Equation 9
Therefore,

76. ACCELERATION—COMPONENTS
Equation 10
Using the formula for curvature given by Theorem 10 in Section 13.3, we have:

77. ACCELERATION—COMPONENTS
Example 7
A particle moves with position function r(t) = ‹t2, t2, t3›
Find the tangential and normal components of acceleration.

78. ACCELERATION—COMPONENTS
Example 7

79. ACCELERATION—COMPONENTS
Example 7
Therefore, Equation 9 gives the tangential component as:

80. ACCELERATION—COMPONENTS
Example 7

81. ACCELERATION—COMPONENTS
Example 7
Hence, Equation 10 gives the normal component as:

82. KEPLER’S LAWS OF PLANETARY MOTION
We now describe one of the great accomplishments of calculus by showing how the material of this chapter can be used to prove Kepler’s laws of planetary motion.

83. KEPLER’S LAWS OF PLANETARY MOTION
After 20 years of studying the astronomical observations of the Danish astronomer Tycho Brahe, the German mathematician and astronomer Johannes Kepler (1571–1630) formulated the following three laws.

84. KEPLER’S FIRST LAW
A planet revolves around the sun in an elliptical orbit with the sun at one focus.

85. KEPLER’S SECOND LAW
The line joining the sun to a planet sweeps out equal areas in equal times.

86. KEPLER’S THIRD LAW
The square of the period of revolution of a planet is proportional to the cube of the length of the major axis of its orbit.

87. KEPLER’S LAWS
In his book Principia Mathematica of 1687, Sir Isaac Newton was able to show that these three laws are consequences of two of his own laws:
Second Law of Motion
Law of Universal Gravitation

88. In what follows, we prove Kepler’s First Law.
The remaining laws are proved as exercises (with hints).

89. KEPLER’S FIRST LAW—PROOF
The gravitational force of the sun on a planet is so much larger than the forces exerted by other celestial bodies.
Thus, we can safely ignore all bodies in the universe except the sun and one planet revolving about it.

90. KEPLER’S FIRST LAW—PROOF
We use a coordinate system with the sun at the origin.
We let r = r(t) be the position vector of the planet.

91. KEPLER’S FIRST LAW—PROOF
Equally well, r could be the position vector of any of:
The moon
A satellite moving around the earth
A comet moving around a star

92. KEPLER’S FIRST LAW—PROOF
The velocity vector is: v = r’
The acceleration vector is: a = r”

93. KEPLER’S FIRST LAW—PROOF
We use the following laws of Newton.
Second Law of Motion: F = ma
Law of Gravitation:

94. KEPLER’S FIRST LAW—PROOF
In the two laws,
F is the gravitational force on the planet
m and M are the masses of the planet and the sun
G is the gravitational constant
r = |r|
u = (1/r)r is the unit vector in the direction of r

95. KEPLER’S FIRST LAW—PROOF
First, we show that the planet moves in one plane.

96. KEPLER’S FIRST LAW—PROOF
By equating the expressions for F in Newton’s two laws, we find that:
So, a is parallel to r.
It follows that r x a = 0.

97. KEPLER’S FIRST LAW—PROOF
We use Formula 5 in Theorem 3 in Section 13.2 to write:

98. KEPLER’S FIRST LAW—PROOF
Therefore, r x v = h
where h is a constant vector.
We may assume that h ≠ 0; that is, r and v are not parallel.

99. KEPLER’S FIRST LAW—PROOF
This means that the vector r = r(t) is perpendicular to h for all values of t.
So, the planet always lies in the plane through the origin perpendicular to h.

100. KEPLER’S FIRST LAW—PROOF
Thus, the orbit of the planet is a plane curve.

101. KEPLER’S FIRST LAW—PROOF
To prove Kepler’s First Law, we rewrite the vector h as follows:

102. KEPLER’S FIRST LAW—PROOF
Then,
(Property 6, Th. 8, Sec. 12.4)

103. KEPLER’S FIRST LAW—PROOF
However, u · u = |u|2 = 1
Also, |u(t)| = 1
It follows from Example 4 in Section that: u · u’ = 0

104. KEPLER’S FIRST LAW—PROOF
Therefore,
Thus,

105. KEPLER’S FIRST LAW—PROOF
Equation 11
Integrating both sides of that equation, we get: where c is a constant vector.

106. KEPLER’S FIRST LAW—PROOF
At this point, it is convenient to choose the coordinate axes so that the standard basis vector k points in the direction of the vector h.
Then, the planet moves in the xy-plane.

107. KEPLER’S FIRST LAW—PROOF
As both v x h and u are perpendicular to h, Equation 11 shows that c lies in the xy-plane.

108. KEPLER’S FIRST LAW—PROOF
This means that we can choose the x- and y-axes so that the vector i lies in the direction of c.

109. KEPLER’S FIRST LAW—PROOF
If θ is the angle between c and r, then (r, θ) are polar coordinates of the planet.

110. KEPLER’S FIRST LAW—PROOF
From Equation 11 we have: where c = |c|.

111. KEPLER’S FIRST LAW—PROOF
Then, where e = c/(GM).

112. KEPLER’S FIRST LAW—PROOF
However,
where h = |h|.

113. KEPLER’S FIRST LAW—PROOF
Thus,

114. KEPLER’S FIRST LAW—PROOF
Equation 12
Writing d = h2/c, we obtain:

115. KEPLER’S FIRST LAW—PROOF
Comparing with Theorem 6 in Section 10.6, we see that Equation 12 is the polar equation of a conic section with:
Focus at the origin
Eccentricity e

116. KEPLER’S FIRST LAW—PROOF
We know that the orbit of a planet is a closed curve.
Hence, the conic must be an ellipse.

117. KEPLER’S FIRST LAW—PROOF
This completes the derivation of Kepler’s First Law.

118. KEPLER’S LAWS
The proofs of the three laws show that the methods of this chapter provide a powerful tool for describing some of the laws of nature.