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1. An object moves along the horizontal coordinate line according to the formula
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where s is the directed distance from the origin in feet
s  t   t 3  20t 2  60t  30 
t
and t > 0 is time in seconds.
• Find the velocity (as a function of time).
• Find the speed (as a function of time).
• Find the acceleration (as a function of time).
• Find all times when the object is moving to the right.
• Find all times when the object’s acceleration is negative.
• Create a well-labeled graph showing the position, velocity, speed, and acceleration on
a window that confirms all of the other results for this problem.
• On the interval 1  t  15 , what is the greatest distance between the object and the
origin? When does this occur?
• When does the object attain its highest velocity on the interval 1  t  15 ? What is
this velocity?
2. Two particles move along a coordinate line. Both objects begin at the origin at time
t  0 . After t seconds (assume t > 0) their directed distances from the origin, in feet,
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are given by s1 
3t 5  8t 4  50t and s2  8t 2  12t  t 3 respectively.
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• When do the objects have the same position?
• When do the objects have the same velocity?
• When do the objects have the same speed?
• In general, which is larger: the number of times two objects have the same velocity or
the number of times two objects have the same speed? (Explain.)
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3. A body moves in a straight line according to this equation of motion:
s(t) = 10t ² − 4t + 8,
where t is measured in seconds and s in meters.
a) What is its position at the end of 5 sec?
b) What is the equation for its velocity v at any time t ?
c) What is its velocity v at at the end of 5 seconds?
d) What is the equation for its acceleration a at any time t ?
e) What is its acceleration at the end of 5 seconds?
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4.
A particle moving along the x -axis has position
after an elapsed time of seconds.
(a) Find the velocity of the particle at time
(b) Find the acceleration at time
(c) What is the total distance travelled by the particle during the first 3 seconds?
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5. A particle is moving along a horizontal line according to s = t − 12t + 36t − 24, t ≥0. Determine
the intervals of time when the particle is moving to the right and when it is moving to the left.
Also determine the instant when the particle reverses its direction.
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6. The position of an object dropped from the Arch is given by s(t) = −16t + 638. If the Arch is
638 feet tall, how long will it take the object to reach the ground? How fast will the object be
going at impact?
7. An object is thrown upward from the ground with an initial velocity of 30 feet per second. Its
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position is given by s(t) = − 16t + 30t. How high will the object go? With what velocity will it
stike the ground?
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8. The position of a stone thrown upward from the ground is given by x(t) = −16t + 32t, where s is
measured in feet and t in seconds. Find (a) the average velocity on the interval (.5,.75), (b)
instantaneous velocity at .5 seconds and .75 seconds, (c) the speed at .5 seconds and .75
seconds, (d) how many seconds it will take the stone to reach the highest point, (e) how high the
stone will go, (f) how many seconds it will take to reach the ground, (g) the instantaneous
velocity of the stone when it hits the ground.
9. A billiard ball is hit and travels in a straight line. If s centimeters is the distance of the ball
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from its initial position at t seconds, then s = 100t + 100t. If the ball hits a cushion that is 39
cm from its initial position, at what velocity does it hit the cushion?
10. A ball is thrown vertically upward from the top of a building 112 feet high. Its position above
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ground level is given by s = −16t + 96t + 112. How high will the ball go? How long will it take to
reach its maximum height? How long will it take the ball to hit the ground? How fast will the
ball be going when it hits the ground?
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11. An object moves on a horizontal coordinate line. Its directed distance, s, from the origin at
the end of t seconds is s  t 3  6t 2  9t feet.
a. When is the object moving to the left?
b. What is its acceleration when its velocity is equal to zero?
c. When is its acceleration positive?
12. A ball is thrown vertically upward from the top of a building 112 ft high with an initial velocity
of 96 ft/second.
a. Find the instantaneous velocity of the ball at t = 2.
b. Find the maximum height that the ball will reach.
c. How long will it take the ball to reach the ground?