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Calculus 1 : Position

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Example Question #1 :How To Find Position

Find a vector perpendicular to (4,3).

Possible Answers:

(3,-4)

(8,6)

(4,-3)

(-4,-3)

(-4,3)

Correct answer:(3,-4)

Explanation:

In general, if we have a vector (a,b), a perpendicular vector is (b,-a).

So here, the perpendicular vector is (3,-4).

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Example Question #1 :How To Find Position

ifa=i+ 2j- 3kandb=4i+ 7k, express the vector 3a+ 2b.

Possible Answers:

14i + 4j + 15k

10i + 5j + 6k

11i + 6j + 5k

12i - 6j + 6k

2i + 4j - 6k

Correct answer:11i + 6j + 5k

Explanation:

To express the vector in terms of i, j, and k, we need to combine like terms and distribute.

3a+ 2b

= 3(i+ 2j- 3k) + 2(4i+ 7k)

= 3i+ 6j- 9k+ 8i+ 14k

= 11i+6j+ 5k

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Example Question #3 :How To Find Position

The velocity of a particle is given by the function $v(t) = 3\sqrt{t}+2t$ . What is it's position at time t = 4 if it's starting position was 4

Possible Answers:

128

Correct answer:

Explanation:

To find the position from velocity, the function must be integrated. This gives $s(t)=2t^{3/2}+t^2+C$ . substituting 4 forand using the given initial condition gives the answer

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Example Question #1 :How To Find Position

The veloctiy of a particle at timeis given by v(t) = 6t . What is its change in position between time t=1 and time t=3 ?

Possible Answers:

Cannot be determined.

Correct answer:

Explanation:

The position function is the intergral of the velocity function. So here, position is given by s = 3t^2+C whereis the constant of integration. Because only a difference in position is asked, and not an absolute position, the constant of integration cancels out.

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Example Question #5 :How To Find Position

Find the position at t=2 if the acceleration function is: a(t) =1 .

Possible Answers:

$\frac{1}{2}$

Correct answer:

Explanation:

To find the position from the acceleration function, integrate the acceleration function twice.

$v(t)= \int a(t)dt = t$

$s(t) = \int v(t) dt = \frac{t^2}{2}$

Substitute t=2 to find the postion.

$s(t=2)= \frac{2^2}{2} = 2$

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Example Question #6 :How To Find Position

Find the position at t=2 if the acceleration is: a(t)=t^2+2t+1 .

Possible Answers:

$\frac{26}{3}$

Correct answer:

Explanation:

To find the position function, integrate the acceleration function twice.

$v(t)= \int a(t) dt= \int (t^2+2t+1 )dt = \frac{1}{3}t^3 + t^2 +t$

$s(t)=\int v(t)dt=\int (\frac{1}{3}t^3+t^2+t) \:dt=\frac{1}{12}t^4+\frac{1}{3}t^3+\frac{1}{2}t^2$

Evalute the position at t=2 .

$s(t=2)=\frac{1}{12}(2)^4+\frac{1}{3}(2)^3+\frac{1}{2}(2)^2=\frac{4}{3}+\frac{8}{3}+2=6$

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Example Question #7 :How To Find Position

The velocity of an object is given by the following equation:

v(t)=7t+4

If s(0)=5 , find the equation for the position of the object at any time.

Possible Answers:

s(t)=7t+9

$s(t)=\frac{7}{2}t^2+4t+5$

s(t)=4t^2+7t+2

s(t)=7t^2+2t+5

$s(t)=\frac{7}{2}t+5$

Correct answer:

$s(t)=\frac{7}{2}t^2+4t+5$

Explanation:

Velocity is the derivative of position, so in order to obtain an equation for position, we must integrate the given equation for velocity:

$s(t)=\int v(t)dt=\int (7t+4)dt=\frac{7}{2}t^2+4t+C$

The next step is to solve for C by applying the given initial condition, s(0)=5:

$s(t)=\frac{7}{2}t^2+4t+C$

$s(0)=0+0+C=5\rightarrow C=5$

So our final equation for position is:

$s(t)=\frac{7}{2}t^2+4t+5$

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Example Question #1 :How To Find Position

The position function of a ball from the ground when it is thrown by a pitcher is p(t)=4t+2 .

Where is the ball located at t=1 ?

Possible Answers:

p(1)=6

p(1)=3

p(1)=2

p(1)=1

p(1)=4

Correct answer:

p(1)=6

Explanation:

To find the position of the ball, we plug in t=1

So p(t)=4t+2 turns into:

$p(1)=4\cdot 1+2$

p(1)=4+2

p(1)=6

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Example Question #9 :How To Find Position

Function v(t) gives the velocity of a particle as a function of time.

$\small \small v(t)=t^5+3t^4-\frac{7t^2}{3}$

Find the equation that models the particle's postion as a function of time.

Possible Answers:

$\small \small h(t)=\frac{t^6}{5}+\frac{3t^5}{4}-\frac{7t^3}{9}+c$

$\small \small h(t)=\frac{t^6}{6}+\frac{3t^5}{7}-\frac{7t^3}{27}+c$

$\small h(t)=\frac{t^6}{6}+\frac{3t^5}{5}-\frac{7t^3}{9}+c$

$\small \small h(t)=\frac{t^6}{6}+\frac{3t^5}{5}+\frac{7t^3}{9}+c$

$\small \small h(t)=\frac{t^6}{6}+\frac{3t^5}{5}-\frac{7t^3}{9}$

Correct answer:

$\small h(t)=\frac{t^6}{6}+\frac{3t^5}{5}-\frac{7t^3}{9}+c$

Explanation:

Recall that velocity is the first derivative of position, and acceleration is the second derivative of position. We begin with velocity, so we need to integrate to find position and derive to find acceleration.

We are starting with the following

$\small \small v(t)=t^5+3t^4-\frac{7t^2}{3}$

We need to perform the following:

$\small \small \small \small\int v(t)dt=\int t^5+3t^4-\frac{7t^2}{3}dt$

Recall that to integrate, we add one to each exponent and divide by the that number, so we get the following. Don't forget your +c as well.

$\small \small \int t^5+3t^4-\frac{7t^2}{3}dt=\frac{t^6}{6}+\frac{3t^5}{5}-\frac{7t^3}{9}+c$

Which makes our position function, h(t), the following:

$\small h(t)=\frac{t^6}{6}+\frac{3t^5}{5}-\frac{7t^3}{9}+c$

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Example Question #10 :How To Find Position

Consider the velocity function modeled in meters per second by v(t).

v(t)=4t^3-5t^2+6

Find the position of a particle whose velocity is modeled by v(t) afterseconds.

Possible Answers:

$93.3\bar{3}+c {m}$

$8393.3\bar{3}+c \text{ }\frac{m}{s}$

$8393.3\bar{3} m$

$393.3\bar{3}m$

$833.3\bar{3}+c m$

Correct answer:

$8393.3\bar{3}+c \text{ }\frac{m}{s}$

Explanation:

Recall that velocity is the first derivative of position, so to find the position function we need to integrate v(t) .

$V(t)=\int 4t^3-5t^2+6dt$

Becomes,

$V(t)=t^4-\frac{5t^3}{3}+6t+c$

Then, we need to find V(10)

$V(10)=10^4-\frac{5\cdot10^3}{3}+6\cdot10+c=8393.3\bar{3}+c$

So our final answer is:

$8393.3\bar{3}+c \text{ }\frac{m}{s}$

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Calculus 1 : Position

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #1 :How To Find Position

Example Question #1 :How To Find Position

Example Question #3 :How To Find Position

Example Question #1 :How To Find Position

Example Question #5 :How To Find Position

Example Question #6 :How To Find Position

Example Question #7 :How To Find Position

Example Question #1 :How To Find Position

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