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Rectilinear MotionRelated Rates
Rectilinear Motion and Related Rates
Mathematics 73Department of Physical Sciences and Mathematics
Rectilinear MotionRelated Rates
Rectilinear Motion or Motion along a Line
Suppose a particle moves along a straight line and its position(relative to the origin) after t units of time is s(t), called theposition function.
Recall that the average velocity of the particle on [t0, t] is
vave =s(t) s(t0)
t t0=
s
t
Rectilinear MotionRelated Rates
Rectilinear Motion or Motion along a Line
Suppose a particle moves along a straight line and its position(relative to the origin) after t units of time is s(t), called theposition function.
Recall that the average velocity of the particle on [t0, t] is
vave =s(t) s(t0)
t t0=
s
t
Rectilinear MotionRelated Rates
Instantaneous Velocity
Definition: The instantaneous velocity of the particle at time tis
v(t) = limt0
s
t= s
(t)
The instantaneous speed of the particle at time t is |v(t)|.
Rectilinear MotionRelated Rates
Instantaneous Velocity
Definition: The instantaneous velocity of the particle at time tis
v(t) = limt0
s
t= s
(t)
The instantaneous speed of the particle at time t is |v(t)|.
Rectilinear MotionRelated Rates
Instantaneous Acceleration
Definition: The instantaneous acceleration of the particle attime t is
a(t) = limt0
v
t= v
(t) = s(t)
Rectilinear MotionRelated Rates
CONVENTION: If the rectilinear motion is horizontal, then thedisplacement is positive if the particle moves to the right; andnegative if it moves to the left of the origin.
Similarly, if the rectilinear motion is vertical, then the displacementis positive if the particle moves upward; and negative if it movesdownward.
Rectilinear MotionRelated Rates
CONVENTION: If the rectilinear motion is horizontal, then thedisplacement is positive if the particle moves to the right; andnegative if it moves to the left of the origin.
Similarly, if the rectilinear motion is vertical, then the displacementis positive if the particle moves upward; and negative if it movesdownward.
Rectilinear MotionRelated Rates
REMARKS: Let s(t) be the position function of a particle movingalong a line. Note that t is nonnegative. The signs of v(t) anda(t) give us information about the motion of the particle.
1 If v(t) > 0, then the particle is moving in the positivedirection of the line at time t.
2 If v(t) < 0, then the particle is moving in the negativedirection of the line at time t.
3 If v(t) = 0, either the particle is not moving or is changingdirection at time t.
Rectilinear MotionRelated Rates
REMARKS: Let s(t) be the position function of a particle movingalong a line. Note that t is nonnegative. The signs of v(t) anda(t) give us information about the motion of the particle.
1 If v(t) > 0, then the particle is moving in the positivedirection of the line at time t.
2 If v(t) < 0, then the particle is moving in the negativedirection of the line at time t.
3 If v(t) = 0, either the particle is not moving or is changingdirection at time t.
Rectilinear MotionRelated Rates
REMARKS: Let s(t) be the position function of a particle movingalong a line. Note that t is nonnegative. The signs of v(t) anda(t) give us information about the motion of the particle.
1 If v(t) > 0, then the particle is moving in the positivedirection of the line at time t.
2 If v(t) < 0, then the particle is moving in the negativedirection of the line at time t.
3 If v(t) = 0, either the particle is not moving or is changingdirection at time t.
Rectilinear MotionRelated Rates
REMARKS: Let s(t) be the position function of a particle movingalong a line. Note that t is nonnegative. The signs of v(t) anda(t) give us information about the motion of the particle.
1 If v(t) > 0, then the particle is moving in the positivedirection of the line at time t.
2 If v(t) < 0, then the particle is moving in the negativedirection of the line at time t.
3 If v(t) = 0, either the particle is not moving or is changingdirection at time t.
Rectilinear MotionRelated Rates
REMARKS:
1 If a(t) > 0, then the velocity of the particle is increasing attime t. In addition,
1 if v(t) > 0, then the speed of the particle is increasing at timet (speeding up).
2 if v(t) < 0, then the speed of the particle is decreasing at timet (speeding down).
2 If a(t) < 0, then the velocity of the particle is decreasing attime t. In addition,
1 if v(t) > 0, then the speed of the particle is decreasing at timet
2 if v(t) < 0, then the speed of the particle is increasing at timet
3 If a(t) = 0, then the velocity of the particle is constant. (Thisdoes not necessarily mean that the particle if not moving.)
Rectilinear MotionRelated Rates
REMARKS:
1 If a(t) > 0, then the velocity of the particle is increasing attime t. In addition,
1 if v(t) > 0, then the speed of the particle is increasing at timet (speeding up).
2 if v(t) < 0, then the speed of the particle is decreasing at timet (speeding down).
2 If a(t) < 0, then the velocity of the particle is decreasing attime t. In addition,
1 if v(t) > 0, then the speed of the particle is decreasing at timet
2 if v(t) < 0, then the speed of the particle is increasing at timet
3 If a(t) = 0, then the velocity of the particle is constant. (Thisdoes not necessarily mean that the particle if not moving.)
Rectilinear MotionRelated Rates
REMARKS:
1 If a(t) > 0, then the velocity of the particle is increasing attime t. In addition,
1 if v(t) > 0, then the speed of the particle is increasing at timet (speeding up).
2 if v(t) < 0, then the speed of the particle is decreasing at timet (speeding down).
2 If a(t) < 0, then the velocity of the particle is decreasing attime t. In addition,
1 if v(t) > 0, then the speed of the particle is decreasing at timet
2 if v(t) < 0, then the speed of the particle is increasing at timet
3 If a(t) = 0, then the velocity of the particle is constant. (Thisdoes not necessarily mean that the particle if not moving.)
Rectilinear MotionRelated Rates
EXAMPLE: A particle moves along the x-axis satisfying the lawof motion s(t) = t3 3t2 + 5, where s is the directed distancefrom the origin. (s is measured in feet and t in seconds)
1 Find the instantaneous velocity and instantaneous accelerationat any time t.
2 Find the time when the particle reverses direction.
3 Find the time when the particle is moving with constantvelocity.
4 Find the velocity and acceleration at t = 4.
5 Show the motion of the particle schematically.
6 Determine the distance traveled by particle during the firstfour seconds.
Rectilinear MotionRelated Rates
EXAMPLE: A particle moves along the x-axis satisfying the lawof motion s(t) = t3 3t2 + 5, where s is the directed distancefrom the origin. (s is measured in feet and t in seconds)
1 Find the instantaneous velocity and instantaneous accelerationat any time t.
2 Find the time when the particle reverses direction.
3 Find the time when the particle is moving with constantvelocity.
4 Find the velocity and acceleration at t = 4.
5 Show the motion of the particle schematically.
6 Determine the distance traveled by particle during the firstfour seconds.
Rectilinear MotionRelated Rates
Remark: For vertical motion, if an object is in free-fall, then theonly force that acts on it during its flight is gravity.
The distance of the object from the ground t seconds after itslaunch is given by
s(t) = h0 + v0t +1
2gt2
where
h0 is the initial height of the object from the ground
v0 is the initial velocity when the object was lauched
g is the acceleration due to gravity, 32 ft/s2
Rectilinear MotionRelated Rates
Remark: For vertical motion, if an object is in free-fall, then theonly force that acts on it during its flight is gravity.
The distance of the object from the ground t seconds after itslaunch is given by
s(t) = h0 + v0t +1
2gt2
where
h0 is the initial height of the object from the ground
v0 is the initial velocity when the object was lauched
g is the acceleration due to gravity, 32 ft/s2
Rectilinear MotionRelated Rates
EXAMPLE: A ball is thrown vertically upward from the groundwith an initial velocity of 32 ft/s.
1 Find the time when the ball reaches the highest point.
2 Determine the highest point reached by the ball.
3 Determine the velocity when t = 0.75 and t = 1.25
4 Determine the speed of the ball when it reaches the ground.
Rectilinear MotionRelated Rates
EXAMPLE: A ball is thrown vertically upward from the groundwith an initial velocity of 32 ft/s.
1 Find the time when the ball reaches the highest point.
2 Determine the highest point reached by the ball.
3 Determine the velocity when t = 0.75 and t = 1.25
4 Determine the speed of the ball when it reaches the ground.
Rectilinear MotionRelated Rates
EXAMPLE: A tomato is thrown upward with an initial velocity of96 ft/s from the top of a building which is 112 ft high. At thesame time, a kitten is walking along the ground along the samehorizontal as the foot of the building according to the equation ofmotion s(t) = t3 18t2 + 96t 133, where s is the distance ofthe kitten from the point on the ground directly under the tomatoat time t.
1 What is the maximum height reached by the tomato?
2 How fast is the tomato going when it hits the ground?
3 In what interval of time is the kitten moving to the left?
4 In what interval of time is the kitten accelarating?
5 Will the tomato hit the kitten? Explain.
Rectilinear MotionRelated Rates
Related Rates
Rectilinear MotionRelated Rates
Suppose x is a quantity that is a function of time t. Thendx
dtis
the rate of change of x with respect to t.
A problem on related rates is a problem involving rates of changeof several variables where one variable is dependent on another. Inparticular, if y is dependent on x , then the rate of change of ywith respect to t is dependent on the rate of change of x with
respect to t, that is,dy
dtis dependent on
dx
dt.
Rectilinear MotionRelated Rates
Suggestions in Solving Problems Involving Related Rates
1 If possible, draw a diagram of the problem that is valid for anytime t > 0.
2 Identify the time-dependent quantities.
3 Write down all numerical facts known about the variables.Interpret each rate of change as the derivative of a variablewith respect to time. If a quantity decreases over time, thenits rate of change is negative.
4 Set-up the equation relating the time-dependent quantities.
5 Differentiate both sides of the equation implicitly with respectto t to relate the time rates.
6 Substitute the values of the known quantities and solve forthe desired quantity.
7 Write a conclusion that answers the question of the problem.
Rectilinear MotionRelated Rates
Example: A ladder 10 meters long is leaning against a wall. Ifthe bottom of the ladder is being pulled horizontally towards thewall at 2 m/s, how fast is the top of the ladder moving when thebottom is 6 meters from the wall?
Rectilinear MotionRelated Rates
Example: Water is pouring into an inverted cone at the rate of 8cubit feet per minute. If the height of the cone is 12 ft and theradius of its base is 6 ft, how fast is the water level rising when thewater is 4 ft deep?
Rectilinear MotionRelated Rates
Related Rates
1 A cubical tank of ice is melting at the rate of 4 cubic metersper minute. At what rate does the side of the ice change atthe instant when the side is 2 meters.
2 A 15-foot ladder is leaning against a vertical wall and ahorizontal floor. The top of the ladder started sliding againstthe wall at the rate of 2 ft/sec. At what rate does the foot ofthe ladder slide along the floor at the instant when the foot ofthe ladder is 12 ft from the wall?
3 A water tank in the form of an inverted cone is being emptiedat the rate of 6 cubic meters per minute. The altitude of thecone is 24 m and the radius is 12 m. Find how fast the waterlevel is lowering when the water is 10 m deep.
Rectilinear MotionRelated Rates
Related Rates
4 A stone is dropped into a still pond. Concentric circle ripplesspread out, and the radius of the disturbed region increases atthe rate of 16 cm/sec. At what rate does the area of thedisturbed region increase when its radius is 4 cm?
5 A boy 4 feet tall is walking towards a 12 feet high lightedlamppost at the rate of 3 feet per second. At what rate doesthe length of his shadow change?
6 Two cars, one going due east at the rate of 90 km/hr and theother going due south at the rate of 60 km/hr, are travelingtoward the intersection of two roads. At what rate are thecars approaching each other at the instant when the first caris 0.2 km and the second car is 0.15 km from the intersection?
Rectilinear MotionRelated Rates
Related Rates
7 An automobile traveling at a rate of 30 ft/sec is approachingan intersection. When the automobile is 120 ft from theintersection, a truck traveling at the rate of 40 ft/sec crossesthe intersection. The automobile and the truck are on roadsthat are at right angles to each other. How fast are theautomobile and the truck separating 2 seconds after the truckleaves the intersection.
8 A baseball diamond is a square whose sides are 90 ft long.Suppose that a player running from the second to the thirdbase has a speed of 30 ft/sec at the instant when he is 20 ftfrom the third base. At what rate is the players distance fromthe home base changing at that instant?
Rectilinear MotionRelated Rates