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Resources

Relate position, velocity, speed, and acceleration using derivatives

s given by my friend pls solve (Im pre foundation so I dont understand shit here)

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20

. The acceleration

a

\text{ms}^{-2}

, of a particle is given by

a=3t^{2}+2t+2

t

is the time. If the particle starts out with a velocity

v=2\text{ms}^{-1}

t=0

, then the velocity at the end of

2\text{s}

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12\text{ms}^{-1}

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14\text{ms}^{-1}

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16\text{ms}^{-1}

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\text{ms}^{-2}

0

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21

. A point Initially at rest moves along X-axis. Its acceleration varies with time as

\text{ms}^{-2}

1

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22

. If the velocity of a particle is

\text{ms}^{-2}

2

\text{ms}^{-2}

3

\text{ms}^{-2}

4

are constants, then the distance travelled by It between is and

2\text{s}

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\text{ms}^{-2}

6

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\text{ms}^{-2}

7

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\text{ms}^{-2}

8

\text{ms}^{-2}

9

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23

. The acceleration of a particle Increasing linearly with time

a=3t^{2}+2t+2

0

. The particle starts from origin with an Initial velocity

a=3t^{2}+2t+2

1

. The distance travelled by the particle In time

t

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a=3t^{2}+2t+2

3

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a=3t^{2}+2t+2

4

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a=3t^{2}+2t+2

5

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a=3t^{2}+2t+2

6

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24

. The velocity of a particle is

a=3t^{2}+2t+2

7

. If its position

a=3t^{2}+2t+2

8

t=0

, then its displacement after unit time

t

0

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t

1

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t

2

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t

3

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t

4

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25

. A particle moving along x-axis has acceleration

a

t

t

7

t

8

t

9

are constants. The particle at

t=0

has zero velocity. In the time interval between

t=0

and the instant when

v=2\text{ms}^{-1}

2

, the particle velocity

v=2\text{ms}^{-1}

3

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v=2\text{ms}^{-1}

4

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v=2\text{ms}^{-1}

5

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v=2\text{ms}^{-1}

6

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v=2\text{ms}^{-1}

7

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26

. A particle located at

v=2\text{ms}^{-1}

8

t=0

, starts moving along the positive x-direction with a velocity

t=0

0

t=0

1

. The displacement of the particle varies with time as

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t=0

2

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t=0

3

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t=0

4

t=0

5

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27

. The velocity of particle moving in the positive direction of X-axis varies as

t=0

6

t=0

7

is a positive constant. Assuming that at moment

t=0

, the particle was located at the point

v=2\text{ms}^{-1}

8

. Find (i) the time dependence of the position (ii) the time dependence of the velocity of the particle. (ii) the mean velocity of the particle averaged over the time that the particle takes to cover first 's' meters of the path [Ans: (i)

2\text{s}

0

2\text{s}

1

2\text{s}

2

The braking distance

D(v)

(in meters) for a certain car moving at velocity

v

(in meters/second) is given by

D(v)=\frac{v^{2}}{22}

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The car's velocity

B(t)

(in meters/second)

t

seconds after starting is given by

B(t)=9 t

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Write a formula for the braking distance

S(t)

(in meters) after

t

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It is not necessary to simplify.

www-awualekscom/alelscgi/X/slexe/to_u-lgNsikasNW

8

8

9

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Craphs and functions

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Word problom involving composition of two functions

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The braking distance

D(v)

(in meters) for a certain car moving at velocity

v

(in meters/second) is given by

D(v)=\frac{v^{2}}{22}

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The car's velocity

B(t)

(in meters/second)

t

seconds after starting is given by

B(t)=9 t

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Write a formula for the braking distance

S(t)

(in meters) after

t

A block with mass

M

moving at a velocity of

V

collides and sticks to a block of mass

2M

initially at rest.What their velocity after the collision?

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\frac{V}{3}

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\frac{V}{2}

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V

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\frac{3V}{2}

A block with mass

M

moving at a velocity of

V

collides and sticks to a block of mass

2M

initially at rest.What their velocity after the collision?

A bloc with mass

M

moving at a velocity

2V

collides elastically with a second identical block moving at a velocity of

V

.What is the velocity of the first block after the collision?

V=\pi r^{2} h

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The equation gives the Volume

V

of a right cylinder with radius

r

h

. Which of the following equations correctly gives the radius of the cylinder in terms of the cylinder's volume and height?

The diagram shows a ballistic pendulum.A

200\,\text{g}

bullet is fires into the suspended

4\,\text{kg}

block of wood and reains embedded inside it (a perfectly inelastic collisions).After the impact of the bullet,the block swings up to a maximum height

h

.The initial speed of the bullet was

5\,\text{m/s}

c.What was the velocity of the bullet block system just after the collision

As a particle moves along the number line, its position at time

t

s(t)

, its velocity is

v(t)

, and its acceleration is

a(t)=-\cos t

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v(\pi)=2

s(\pi / 2)=3 \pi

s(0)

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s(0)=

The position of a particle moving in the

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xy

-plane is given by the parametric equations

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x(t)=\frac{6t}{t+1}

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y(t)=\frac{-8}{t^{2}+4}

. What is the slope of the line tangent to the path of the particle at the point where

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t=2

70\,\text{kg}

is placed on the floor of a lift of mass

630\,\text{kg}

if the lift is moving downwards with uniform acceleration

1.4

then the magnitiude of the ttension in the string which carries the lift equals ..

\text{kg.w}

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AP Exam Review Free Response

4

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This Question is

{ }^{* *}

CALCULATOR ACTIVE

{ }^{* *}

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Please show all work on page

2 \& 3

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t \geq 0

, a particle moves along the

x

-axis. The velocity of the particle at time

t

v(t)=1-\tan ^{-1}(2 t)

. The particle is at position

x=-4

t=0

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t=3

, what is the acceleration of the particle?

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t=3

, is the speed of the particle increasing or decreasing? Give a reason for your answer.

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t \geq 0

, find the time when the particle is farthest to the right. Justify your answer.

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(d) Find the position of the particle at time

t=3

. Is the particle moving toward the origin or away from the origin at

t=3

? Justify your answer.

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AP Exam Review Free Response

4

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This Question is **CALCULATOR ACTIVE**

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Please show all work on page

2 \& 3

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t \geq 0

, a particle moves along the x-axis. The velocity of the particle at time

t

v(t)=1-\tan ^{-1}(2 t)

. The particle is at position

x=-4

t=0

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t=3

, what is the acceleration of the particle?

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t=3

, is the speed of the particle increasing or decreasing? Give a reason for your answer.

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t \geq 0

, find the time when the particle is farthest to the right. Justify your answer.

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(d) Find the position of the particle at time

t=3

. Is the particle moving toward the origin or away from the origin at

t=3

? Justify your answer.

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- A particle moves along a horizontal line so that its position is given by

s(t)=t^{3}-9 t^{2}+15 t+4

t \geq 0

. When is the particle speeding up?

Below is the graph of a trigonometric function. It intersects its midline at

(\pi\tau, 2.75)

, and it has a maximum point at

(\frac{5}{4}\pi\tau, 6.75)

what is the amplitude of the function

Below is the graph of a trigonometric function. It intersects its midline at

(\pi\tau, 2.75)

, and it has a maximum point at

(\frac{5}{4}\pi\tau, 6.75)

what is the amplitude of the function

An object travels along a straight line. The function

v(t) = -13t^3 - 2t^2 + 8t + 9

gives the object's velocity, in feet per second, at time

t

seconds. Write a function that gives the object's acceleration

a(t)

in feet per second per second.

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An object travels along a straight line. The function

s(t) = 5t^3 + 12t^2 - 15t - 10

gives the object's position, in feet, at time

t

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Write a function that gives the object's velocity

v(t)

in feet per second.

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v(t) =

An object travels along a straight line. The function

s(t) = -32t^2 - 6t + 45

gives the object's position, in feet, at time

t

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Write a function that gives the object's velocity

v(t)

in feet per second.

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An object travels along a straight line. The function

s(t) = 10t - 3

gives the object's position, in kilometers, at time

t > 0

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Write a function that gives the object's velocity

v(t)

in kilometers per hour.

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v(t) = \underline{\hspace{3cm}}

An object travels along a straight line. The function

v(t) = - t^{\frac{1}{5}} + 2t^{-2}

gives the object's velocity, in feet per minute, at time

t > 0

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Write a function that gives the object's acceleration

a(t)

in feet per minute per minute.

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An object travels along a straight line. The function

s(t) = 3\cos t - 7e^t

gives the object's position, in feet, at time

t

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Write a function that gives the object's velocity

v(t)

in feet per second.

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An object travels along a straight line. The function

s(t) = 3t^{\frac{4}{3}} + 6t^{\frac{1}{3}}

gives the object's position, in feet, at time

t

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Write a function that gives the object's velocity

v(t)

in feet per minute.

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v(t) =

An object travels along a straight line. The function

v(t) = 2\sqrt{t} - 6\sqrt[3]{t}

gives the object's velocity, in meters per second, at time

t

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Write a function that gives the object's acceleration

a(t)

in meters per second per second.

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An object travels along a straight line. The function

v(t) = 5\cos t - 4t

gives the object's velocity, in feet per hour, at time

t

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Write a function that gives the object's acceleration

a(t)

in feet per hour per hour.

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An object travels along a straight line. The function

s(t) = -9.8t^2 + 18t + 100

gives the object's position, in feet, at time

t

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Write a function that gives the object's velocity

v(t)

in feet per second.

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An object travels along a straight line. The function

v(t) = 4\sqrt{t} - 3

gives the object's velocity, in kilometers per hour, at time

t > 0

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Write a function that gives the object's acceleration

a(t)

in kilometers per hour per hour.

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An object travels along a straight line. The function

v(t) = 4t^4 - 3t^3 - 11t + 3

gives the object's velocity, in feet per hour, at time

t

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Write a function that gives the object's acceleration

a(t)

in feet per hour per hour.

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An object travels along a straight line. The function

v(t) = 4t^3 - 12t^2 + 5t + 10

gives the object's velocity, in miles per hour, at time

t

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Write a function that gives the object's acceleration

a(t)

in miles per hour per hour.

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An object travels along a straight line. The function

s(t) = -4t^5 + 10t^4 + 7t^3 + t + 11

gives the object's position, in meters, at time

t

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Write a function that gives the object's velocity

v(t)

in meters per second.

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v(t) =

An object travels along a straight line. The function

v(t) = 3\sqrt{t} - 4t^2 + t

gives the object's velocity, in feet per hour, at time

t

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Write a function that gives the object's acceleration

a(t)

in feet per hour per hour.

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An object travels along a straight line. The function

s(t) = t^2 - 10t + 12

gives the object's position, in miles, at time

t

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Write a function that gives the object's velocity

v(t)

in miles per hour.

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Madison started hiking at an elevation of

\mathbf{4}

meters above sea level. She hiked up at a rate of

\mathbf{3}

meters per minute. The graph below represents the relationship between the number of minutes and the elevation.

4

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What is the slope of the line?

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=

In a sinusoidal wave, the time required for a particular point to move from maximum displacement to zero displacement is

0.170\,\text{s}

. The frequency of the wave is

The potential energy of a particle varies the distance

x

from a fixed origin as

U= \frac{A}{x^2} +B

A

B

are dimensional constants, then find the dimensional formula for

\frac{A^2}{B^2}.

If the velocity of a particle moving along the

x

v(t)=2 t-4

t=0

its position is

4

, then at any time

t

x(t)

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t^{2}-4 t

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t^{2}-4 t-4

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t^{2}-4 t+4

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2 t^{2}-4 t

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2 t^{2}-4 t+4

1

. A jeeprey acceleration of a speed of

40 \mathrm{~m} / \mathrm{s}

over a distance of

300 \mathrm{~m}

. Determine the aceleration (assume uniformly) af the jeeprey.

3

. The distance a person can see through a particular submarine periscope is given by the equation

d=6 \sqrt{h-4}+3

h

is the height in feet above water.

A cheetah accelerating uniformly from rest reaches a speed of

29\,\text{ms}^{-1}

2.0\,\text{s}

and then maintains this speed for

15\,\text{seconds}

. (a) Calculate (i) its acceleration (ii) the distance it travels while accelerating

3

. A block with a mass of

30 \mathrm{~kg}

is located on a horizontal, frictionless tabletop. This block is connected by a rope to another block with a mass of

10 \mathrm{~kg}

. The rope is looped through a pulley on the table's edge so that the less massive block is hanging over the edge. What is the magnitude of the acceleration of the larger block across the table?

14

. The equation for wind speed

w

, in miles per hour, near the center of a tornado is

w=93 \log _{10}(d)+65

d

is the distance in miles that the tornado travels.

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A. Write this equation in exponential form.

P

moves in a straight line with displacement relative to origin given by

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s=2 \sin (\pi t)+\sin (2 \pi t), t \geq 0,

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t

is the time in seconds and the displacement is measured in centimetres.

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(i) Write down the period of the function

s

A particle moves along the

x

-axis with velocity

v(t)=\cos \left(t^{2}-3 t\right)

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What is the total distance the particle has traveled between the times

t=3

t=6

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Use a graphing calculator and round your answer to three decimal places.

A particle moves along the

x

-axis with velocity

v(t)=\frac{t^{2}}{\sin ^{2}(t)+2}

. The particle is at position

x=3

t=2

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What is the particle's position at time

t=7

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Use a graphing calculator and round your answer to three decimal places.

A particle moves along the

x

-axis with velocity

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v(t)=\ln \left(t^{2}+5 t+1\right) \text {. }

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What is the particle's displacement between the times

t=1

t=5

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Use a graphing calculator and round your answer to three decimal places.

Herman is riding his hoverboard. The function

V

gives Herman's velocity (in meters per second),

t

seconds after he started riding.

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What is the best interpretation for the following statement?

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The value of the derivative of

V

t=5

1

5

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1

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A) During the first

5

seconds, Herman's acceleration is

1

5

meters per second squared.

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5

seconds, Herman's acceleration is

1

5

meters per second squared.

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5

seconds, Herman's velocity is

1

5

meters per second.

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5

seconds, Herman's acceleration is

1

5

x=v_{\circ} t+\frac{1}{2} a t^{2}

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The horizontal displacement,

x

, of an object with constant acceleration,

a

, initial velocity,

v_{\circ}

, at elapsed time,

t

, is given by the equation. Which of the following equations correctly shows the acceleration in terms of displacement, initial velocity, and time?

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1

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a=\frac{2 \sqrt{x-v_{\circ} t}}{t}

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a=\frac{2\left(x-v_{0}\right)}{t}

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a=\frac{2\left(x-v_{\circ} t\right)}{t^{2}}

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a=\frac{2 x}{v_{0} t^{3}}

A big ship drops its anchor.

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E

represents the anchor's elevation relative to the water's surface (in meters) as a function of time

t

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E=-2.4 t+75

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How far does the anchor drop every

5

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\square

A tug is moxing hack and forth on a straight path. The velocity of the bug is given by

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v(t)=t^{2}-3t

. Find the average acceleration of the bug on the interval

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[1,4]

A hug is mowing hack and forth on a straight path. The velocity of the bug is given by

v(t)=t^{2}-3 t

. Find the average acceleration of the bug on the interval

[1,4]

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