Resources

Resources

Relate position, velocity, speed, and acceleration using derivatives

The precentege error in a measur - ment of mass and velocity are mesured

3 \%

5 \%

respectively than find

\%

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P=m v)

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b) Kinetic energy

\left(K=\frac{1}{2} m v^{2}\right)

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

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) = 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) = 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) = 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

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) = 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) = -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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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

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) = 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) = 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

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

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) = 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) = -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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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

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

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

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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}}

According to Newton's Second Law of Motion, the sum of the forces that act on an object with a mass

m

that moves with an acceleration

a

ma

. An object whose mass is

80

grams has an acceleration of

20

meters per seconds squared. What calculation will give us the sum of the forces that act on the object, in Newtons (which are

\text{kg} \times \text{m/s}^2

Suppose you are driving a car along a straight road. Your velocity at any time

t

(in seconds) is given by the function

v(t)=3 t^{2}-2 t+5

meters per second. Find the total distance traveled by the car from

t=0

t=3

Force equals mass times acceleration.

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\mathrm{F}=\mathrm{ma}

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Solve the equation for the mass,

m

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\mathrm{m}=\frac{[\square]}{[\square]}

A roller coaster is moving at

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

at the bottom of a hill. Three seconds later it reaches the top of the hill moving at

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

. What was the acceleration of the coaster?

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)=3 t^{2}

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v(0)=3

s(0)=1

s(2) ?

You drop an object off the roof of a

441

-foot building. The function

h(t)=-16 t 2+441

gives the object height

h

in feet above the ground after

t

seconds. After how many seconds does the object hit the ground?

A train travelling at a uniform speed for

360 \mathrm{~km}

would have taken

48

minutes less to travel the same distance if its speed were

5 \mathrm{~km} /

hour more. Find the original speed of the train.

A body constrained to move along the

z

-axis of a coordinate system is subject to a constant force

F

F=-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}} N

\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}

are unit vectors along the

x, y

z

axis of the system respectively. What is the work done by this force in moving the body a distance of

4 \mathrm{~m}

\mathrm{z}

A particle moves along the

x

-axis with velocity given by

v(t)=12 \sin (4 t)

t \geq 0

. If the particle is at position

x=1

t=0

, what is the position of the particle at time

t=\frac{\pi}{8}

T

(in seconds) of a pendulum is given by

T=2\pi\sqrt{(\frac{L}{32})}

L

stands for the length (in feet) of the pendulum. If

\pi=3.14

, and the period is

9.42

seconds, what is the length? The length of the pendulum is in feet.

A cart rolling down an incline for

5.0

seconds has an acceleration of

4.0\,\text{m/s}^{2}

. If the cart has a beginning speed of

2.0\,\text{m/s}

, what is its final speed?

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22\,\text{m/s}

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20\,\text{m/s}

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18\,\text{m/s}

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24\,\text{m/s}

A roller coasters velocity at the top of the hill is

10\frac{m}{s}

. Two seconds later it reaches the bottom of the hill with a velocity of

26\frac{m}{s}

. What is the acceleration of the coaster?

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18\frac{m}{s^{2}}

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10\frac{m}{s^{2}}

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8\frac{m}{s^{2}}

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32\frac{m}{s^{2}}

A particle moves along the

x

-axis so that at time

t \geq 0

its velocity is given by

v(t)=6 t^{2}-24 t+18

. Determine all intervals when the particle is moving to the right.

A particle moves along the

x

-axis so that at time

t \geq 0

its position is given by

x(t)=t^{5}-5 t^{4}

. Determine all intervals when the particle is moving to the right.

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\text { Moving right } \rightarrow v(t)>0

A particle moves along the

x

-axis so that at time

t \geq 0

its position is given by

x(t)=-5 t^{4}+30 t^{2}

. Determine all intervals when the speed of the particle is increasing.

A particle moves along the

x

-axis so that at time

t \geq 0

its position is given by

x(t)=-5 t^{4}+30 t^{2}

. Determine all intervals when the speed of the particle is increasing.

A man flies a kite with a

100

foot string. The angle of elevation of the string is

52^{\circ}

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What is the height of the kite in the sky?

A particle moves along the

x

-axis so that at time

t \geq 0

its velocity is given by

v(t)=18 t^{2}-72 t+54

. Determine all intervals when the speed of the particle is increasing.

A particle moves along the

x

-axis so that at time

t \geq 0

its velocity is given by

v(t)=6 t^{2}-48 t+90

. Determine all intervals when the speed of the particle is decreasing.

A particle moves along the

x

-axis so that at time

t \geq 0

its position is given by

x(t)=4 t^{3}-36 t^{2}

. Determine all intervals when the speed of the particle is increasing.

A particle moves along the

x

-axis so that at time

t \geq 0

its velocity is given by

v(t)=3 t^{2}-24 t+36

. Determine all intervals when the acceleration of the particle is positive.

A particle moves along the

x

-axis so that at time

t \geq 0

its velocity is given by

v(t)=-2 t+6

. Determine if the particle is moving to the right or to the left at

t=10

A particle moves along the

x

-axis so that at time

t \geq 0

its velocity is given by

v(t)=-2 t+6

. Determine if the particle is moving to the right or to the left at

t=10

A particle moves along the

x

-axis so that at time

t \geq 0

its position is given by

x(t)=t^{3}+5 t^{2}-32 t

. Determine if the particle is speeding up or slowing down at

t=1

A small mass attached to a spring oscillates in simple harmonic motion with an amplitude of

50 \mathrm{~mm}

30

seconds to complete

10

oscillations. Calculate its period, angular velocity, maximum speed and the maximum acceleration.