Question

The function

sequals=?f(t)

gives the position of an object moving along the? s-axis as a function of time t. Graph f together with the velocity function

?v(t)equals=StartFraction ds Over dt EndFractiondsdtequals=f prime left parenthesis t right parenthesisf?(t)

and the acceleration function

?a(t)equals=StartFraction d squared s Over dt squared EndFractiond2sdt2equals=f prime prime left parenthesis t right parenthesisf??(t)?,

then complete parts? (a) through? (f).

sequals=112112tminus?16 t squared16t2?,

0less than or equals?tless than or equals?77

?(a heavy object fired straight up from? Earth's surface at 112 ?ft/sec)

?v(t)equals=?

?a(t)equals=?

a. When is the object momentarily at? rest? Select the correct answer? below, and if? necessary, fill in the answer? box(es) to complete your choice.

A.

The object is at rest when

?v(t)equals=nothing

?ft/sec, and this occurs at

tequals=nothing

sec.

?(Type integers or decimals. Use a comma to separate answers as? needed.)

B.

The object is never at rest.

b. When does it move to down or? up? Select the correct answer? below, and if? necessary, fill in the answer? box(es) to complete your choice.

?(Simplify your answer. Type your answer in interval notation. Use integers or decimals for any numbers in the? expression.)

A.

The object is moving down for t in the interval

nothing ?,

but is never moving up.

B.

The object is moving down for t in the interval

nothing

and the object is moving up for t in the interval

nothing .

C.

The object is never moving? down, but is moving up for t in the interval

nothing .

D.

The object is never moving down or up.

c. When does the object change? direction? Select the correct answer? below, and if? necessary, fill in the answer box to complete your choice.

A.

The object changes direction at

tequals=nothing

sec.

?(Type an integer or a decimal. Use a comma to separate answers as? needed.)

B.

The object never changes direction.

d. When does the object speed up and slow? down? Select the correct answer? below, and if? necessary, fill in the answer? box(es) to complete your choice.

?(Simplify your answer. Type your answer in interval notation. Use integers or decimals for any numbers in the? expression.)

A.

The object never speeds? up, but slows down for t in the interval

nothing .

B.

The object speeds up for t in the interval

nothing

and the object slows down for t in the interval

nothing .

C.

The object speeds up for t in the interval

nothing ?,

but never slows down.

D.

The object never speeds up or slows down.

e. When is the object moving fastest? (highest speed)?? Slowest? Select the correct answer below and fill in any answer boxes to complete your choice.

A.

The object is moving fastest at

tequals=nothing ?,

when the speed is

nothing

?ft/sec. The object is moving slowest at

tequals=nothing ?,

when the speed is

nothing

?ft/sec.

?(Type an integer or a decimal. Use a comma to separate answers as? needed.)

B.

The object is moving at a constant speed of ? ?ft/sec.

f. When is the object farthest from the axis? origin?

When tequals=? ?sec, the object is ? feet from the? origin, which is the farthest away.

Answer 1

given

s = f(t)

v(t) = ds/dt = f'(t)

a(t) = d^2s/dt^2 = f"(t)

a. f = 112t - 16t^2

0 < t < 7

the plot is as under

f = 112t - 16t^2

f' = 112 - 32t

plot

b. hevy object fired straight from earth

vo = 112 ft/s

hence

v(t) = vo + a*t

and

a = -g ( acceleration due to gravity)

hence

v(t) = 112 - 32t

s(t) = 112t - 16t^2

a(t) = -32 ft/s/s

a. when the object is at momentary rest

then v(t) = 0

this occurs at 112 = 32t

t = 3.5 s

object changes direction at t = 3.5 s

the object speeds down for 0 < t < 3.5 and speeds up for rest of the interval

the object moves the slowest when t = 3.5 s and v(t0 = 0 m/s)

also the object is farthest form the origin at t = 3.5 s