Last time acceleration as a function of time

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• Last time acceleration as a function of time
• Now acceleration as a function of position
• v dx/dt ? dt dx/v and a dv/dt
• a dv/(dx/v) ? adx vdv

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• Last time acceleration as a function of time
• Now acceleration as a function of position
• v dx/dt ? dt dx/v and a dv/dt
• a dv/(dx/v) ? adx vdv
• 11.29 The acceleration due to gravity of a
  particle falling toward the earth is
• a -gR2/r2, where r is the distance from the
  center of the earth to the particle.
• R is the radius of the earth. If R 3960 mi.,
  calculate the escape velocity, that
• is the minimum velocity that an object must be
  projected upward not to return
• to the earth. (Hint v 0 for r ? )
• f(r) -gR2/r2

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• Acceleration as a function of velocity
• f(v) vdv/dx ? dx vdv/f(v)
• f(v) dv/dt ? dt dv/f(v)
• Example 11.25 The aceleration of a particle is
  defined by the relation
• a -kv2.5, where k is a constant. The particle
  starts at x 0 with a
• velocity of 16 mm/s, and when x 6 mm the
  velocity is observed to be
• 4 mm/s. Determine (a) the velocity of the
  particle when x 5mm, (b)
• The time at which the velocity of the particle is
  9 mm/s.
• Solution part a.

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• Acceleration as a function of velocity
• f(v) vdv/dx ? dx vdv/f(v)
• f(v) dv/dt ? dt dv/f(v)
• Example 11.25 The aceleration of a particle is
  defined by the relation
• a -kv2.5, where k is a constant. The particle
  starts at x 0 with a
• velocity of 16 mm/s, and when x 6 mm the
  velocity is observed to be
• 4 mm/s. Determine (a) the velocity of the
  particle when x 5mm, (b)
• The time at which the velocity of the particle is
  9 mm/s.
• Solution part a.

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• Acceleration as a function of velocity
• f(v) vdv/dx ? dx vdv/f(v)
• f(v) dv/dt ? dt dv/f(v)
• Example 11.25 The aceleration of a particle is
  defined by the relation
• a -kv2.5, where k is a constant. The particle
  starts at x 0 with a
• velocity of 16 mm/s, and when x 6 mm the
  velocity is observed to be
• 4 mm/s. Determine (a) the velocity of the
  particle when x 5mm, (b)
• The time at which the velocity of the particle is
  9 mm/s.
• Solution part a.

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• Acceleration as a function of velocity
• f(v) vdv/dx ? dx vdv/f(v)
• f(v) dv/dt ? dt dv/f(v)
• Example 11.25 The aceleration of a particle is
  defined by the relation
• a -kv2.5, where k is a constant. The particle
  starts at x 0 with a
• velocity of 16 mm/s, and when x 6 mm the
  velocity is observed to be
• 4 mm/s. Determine (a) the velocity of the
  particle when x 5mm, (b)
• The time at which the velocity of the particle is
  9 mm/s.
• Solution part a.

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• Solution part b

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• Solution part b

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• Motion under Constant acceleration
• dv/dt a constant then the equations can be
  integrated to yield
• v v0 at
• x x0 v0t at2/2
• v2 v02 2a(x x0)

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• Motion under Constant acceleration
• dv/dt a constant then the equations can be
  integrated to yield
• v v0 at
• x x0 v0t at2/2
• v2 v02 2a(x x0)
• Problem 11.35 A truck travels 540 ft in 8 s while
  being decelerated at a
• Constant rate of 1.5 ft/s2. Determine its intial
  velocity, its final velocity,
• And the distance traveled during the first 0.6 s.

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• Motion under Constant acceleration
• dv/dt a constant then the equations can be
  integrated to yield
• v v0 at
• x x0 v0t at2/2
• v2 v02 2a(x x0)
• Problem 11.35 A truck travels 540 ft in 8 s while
  being decelerated at a
• Constant rate of 1.5 ft/s2. Determine its intial
  velocity, its final velocity,
• And the distance traveled during the first 0.6 s.
• Solution given a -1.5 ft/s2, x0 0, xf 540
  ft unknown v0 and vf

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• Motion under Constant acceleration
• dv/dt a constant then the equations can be
  integrated to yield
• v v0 at
• x x0 v0t at2/2
• v2 v02 2a(x x0)
• Problem 11.35 A truck travels 540 ft in 8 s while
  being decelerated at a
• Constant rate of 1.5 ft/s2. Determine its intial
  velocity, its final velocity,
• And the distance traveled during the first 0.6 s.
• Solution given a -1.5 ft/s2, x0 0, xf 540
  ft unknown v0 and vf
• x x0 v0t at2/2
• 540 0 v0(8) -1.5(8)2/2 ? v0 73.5 ft/s

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• Motion under Constant acceleration
• dv/dt a constant then the equations can be
  integrated to yield
• v v0 at
• x x0 v0t at2/2
• v2 v02 2a(x x0)
• Problem 11.35 A truck travels 540 ft in 8 s while
  being decelerated at a
• Constant rate of 1.5 ft/s2. Determine its intial
  velocity, its final velocity,
• And the distance traveled during the first 0.6 s.
• Solution given a -1.5 ft/s2, x0 0, xf 540
  ft unknown v0 and vf
• x x0 v0t at2/2
• 540 0 v0(8) -1.5(8)2/2 ? v0 73.5 ft/s
• v v0 at
• vf 73.5 -1.5(8) ? vf 61.5 ft/s

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• Motion under Constant acceleration
• dv/dt a constant then the equations can be
  integrated to yield
• v v0 at
• x x0 v0t at2/2
• v2 v02 2a(x x0)
• Problem 11.35 A truck travels 540 ft in 8 s while
  being decelerated at a
• Constant rate of 1.5 ft/s2. Determine its intial
  velocity, its final velocity,
• And the distance traveled during the first 0.6 s.
• Solution given a -1.5 ft/s2, x0 0, xf 540
  ft unknown v0 and vf
• x x0 v0t at2/2
• 540 0 v0(8) -1.5(8)2/2 ? v0 73.5 ft/s
• v v0 at
• vf 73.5 -1.5(8) ? vf 61.5 ft/s
• At t 0.6 s
• x x0 v0t at2/2
• x 0 73.5(0.6) -1.5(0.6)2/2 ? 43.8 ft

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• Relative motion between two particles
• xB/A xB xA or xB xA xB/A
• vB vA vB/A
• aB aA aB/A

A
B
xA
XB/A
XB
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• Problem 11.40 Two rockets are launched at a
  fireworks performance.
• Rocket A is launched with an initial velocity v0
  and rocket B is launched
• 4 s later with the same initial velocity. The
  two rockets are timed to
• Explode simultaneously at a height of 80 m, as A
  is falling and B is
• Rising. Assuming constant acceleration, g 9.81
  m/s2, determine (a)
• The initial velocity v0, (b) the velocity of B
  relative to A at the time of
• The explosion.
• Solution part a x x0 v0t at2/2 ? for A 80
  0 v0t -gt2/2
• for B 80 0 v0(t-4) -g(t-4)2/2

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• Problem 11.40 Two rockets are launched at a
  fireworks performance.
• Rocket A is launched with an initial velocity v0
  and rocket B is launched
• 4 s later with the same initial velocity. The
  two rockets are timed to
• Explode simultaneously at a height of 80 m, as A
  is falling and B is
• Rising. Assuming constant acceleration, g 9.81
  m/s2, determine (a)
• The initial velocity v0, (b) the velocity of B
  relative to A at the time of
• The explosion.
• Solution part a x x0 v0t at2/2 ? for A 80
  0 v0t -gt2/2
• for B 80 0 v0(t-4) -g(t-4)2/2
• V0 g(t-2) ? 80 0 g(t-2)t -gt2/2 ? t
  8.019 s

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• Problem 11.40 Two rockets are launched at a
  fireworks performance.
• Rocket A is launched with an initial velocity v0
  and rocket B is launched
• 4 s later with the same initial velocity. The
  two rockets are timed to
• Explode simultaneously at a height of 80 m, as A
  is falling and B is
• Rising. Assuming constant acceleration, g 9.81
  m/s2, determine (a)
• The initial velocity v0, (b) the velocity of B
  relative to A at the time of
• The explosion.
• Solution part a x x0 v0t at2/2 ? for A 80
  0 v0t -gt2/2
• for B 80 0 v0(t-4) -g(t-4)2/2
• V0 g(t-2) ? 80 0 g(t-2)t -gt2/2 ? t
  8.019 s
• V0 9.81(8.019-4) ? 39.4 m/s upward

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• Problem 11.40 Two rockets are launched at a
  fireworks performance.
• Rocket A is launched with an initial velocity v0
  and rocket B is launched
• 4 s later with the same initial velocity. The
  two rockets are timed to
• Explode simultaneously at a height of 80 m, as A
  is falling and B is
• Rising. Assuming constant acceleration, g 9.81
  m/s2, determine (a)
• The initial velocity v0, (b) the velocity of B
  relative to A at the time of
• The explosion.
• Solution part a x x0 v0t at2/2 ? for A 80
  0 v0t -gt2/2
• for B 80 0 v0(t-4) -g(t-4)2/2
• V0 g(t-2) ? 80 0 g(t-2)t -gt2/2 ? t
  8.019 s
• V0 9.81(8.019-4) ? 39.4 m/s upward
• part b vB/A vB vA

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• Problem 11.40 Two rockets are launched at a
  fireworks performance.
• Rocket A is launched with an initial velocity v0
  and rocket B is launched
• 4 s later with the same initial velocity. The
  two rockets are timed to
• Explode simultaneously at a height of 80 m, as A
  is falling and B is
• Rising. Assuming constant acceleration, g 9.81
  m/s2, determine (a)
• The initial velocity v0, (b) the velocity of B
  relative to A at the time of
• The explosion.
• Solution part a x x0 v0t at2/2 ? for A 80
  0 v0t -gt2/2
• for B 80 0 v0(t-4) -g(t-4)2/2
• V0 g(t-2) ? 80 0 g(t-2)t -gt2/2 ? t
  8.019 s
• V0 9.81(8.019-4) ? 39.4 m/s upward
• part b vB/A vB vA ? v0 -g(t-4) (v0 gt)

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• Problem 11.40 Two rockets are launched at a
  fireworks performance.
• Rocket A is launched with an initial velocity v0
  and rocket B is launched
• 4 s later with the same initial velocity. The
  two rockets are timed to
• Explode simultaneously at a height of 80 m, as A
  is falling and B is
• Rising. Assuming constant acceleration, g 9.81
  m/s2, determine (a)
• The initial velocity v0, (b) the velocity of B
  relative to A at the time of
• The explosion.
• Solution part a x x0 v0t at2/2 ? for A 80
  0 v0t -gt2/2
• for B 80 0 v0(t-4) -g(t-4)2/2
• V0 g(t-2) ? 80 0 g(t-2)t -gt2/2 ? t
  8.019 s
• V0 9.81(8.019-4) ? 39.4 m/s upward
• part b vB/A vB vA ? v0 -g(t-4) (v0 gt)
  ? 39.3 m/s upward

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• Dependent motion objects connected by inelastic
  cords, cables, belts, or chains.
• Total length of cord
• L xA k1 2xB k2 xC k3

Datum
x
xB
xA
A
xC
B
C
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• Dependent motion objects connected by inelastic
  cords, cables, belts, or chains.
• Total length of cord
• L xA k1 2xB k2 xC k3
• dL/dt dxA/dt 2dxB/dt dxC/dt

Datum
x
xB
xA
A
xC
B
C
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• Dependent motion objects connected by inelastic
  cords, cables, belts, or chains.
• Total length of cord
• L xA k1 2xB k2 xC k3
• dL/dt dxA/dt 2dxB/dt dxC/dt
• 0 vA 2vB vC
• 0 aA 2aB aC

Datum
x
xB
xA
A
xC
B
C
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• Dependent motion objects connected by inelastic
  cords, cables, belts, or chains.
• Total length of cord
• L xA k1 2xB k2 xC k3
• dL/dt dxA/dt 2dxB/dt dxC/dt
• 0 vA 2vB vC
• 0 aA 2aB aC
• vB vA vB/A
• vC vA vC/A

Datum
x
xB
xA
A
xC
B
C
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• Dependent motion multiple cords
• Total length of cord
• L1 xA 2xD

Datum
x
xD
xA
D
A
xC
xB
C
B
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• Dependent motion multiple cords
• Total length of cord
• L1 xA 2xD
• L2 (xB xD) (xC xD)

Datum
x
xD
xA
D
A
xC
xB
C
B
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• Dependent motion multiple cords
• Total length of cord
• L1 xA 2xD
• L2 (xB xD) (xC xD)
• 0 vA 2vD
• 0 (vB vD) (vC vD)
• 2vD vB vC
• 0 vA vB vC

Datum
x
xD
xA
D
A
xC
xB
C
B