Calculus 2 : Applications in Physics

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #196 : Integrals

The velocity of a ship is defined as  (where time  is measured in seconds). What distance (in meters) does the ship travel between  seconds and  seconds?

Possible Answers:

Correct answer:

Explanation:

We define velocity as the derivative of distance, or .

Thus, in order to find the distance traveled when given the velocity, we'll need to take the definite integral of the velocity function over a specified time period, or .

Since , we can use the Power Rule for Integrals

for all ,

to find:

Since the definite integral at  is , we get:

 

 

Example Question #197 : Integrals

The velocity of a car is defined as  (where time  is measured in seconds). What distance (in meters) does the car travel between  seconds and  seconds?

Possible Answers:

Correct answer:

Explanation:

We define velocity as the derivative of distance, or .

Thus, in order to find the distance traveled when given the velocity, we'll need to take the definite integral of the velocity function over a specified time period, or .

Since , we can use the Power Rule for Integrals

 for all ,

to find:

Since the definite integral at  is , we get:

Example Question #198 : Integrals

The velocity of a rocket is defined as  (where time  is measured in seconds). What distance (in meters) does the rocket travel between  second and  seconds?

Possible Answers:

Correct answer:

Explanation:

We define velocity as the derivative of distance, or .

Thus, in order to find the distance traveled when given the velocity, we'll need to take the definite integral of the velocity function over a specified time period, or .

Since , we can use the Power Rule for Integrals

 for all ,

to find:

Example Question #199 : Integrals

A dog travels a certain distance between  seconds and  seconds. If we define its velocity as , what is that distance in meters?

Possible Answers:

Correct answer:

Explanation:

We define velocity as the derivative of distance, or .

Thus, in order to find the distance traveled when given the velocity, we'll need to take the definite integral of the velocity function over a specified time period, or .

Since , we can use the Power Rule for Integrals

 for all ,

to find:

Example Question #200 : Integrals

A skateboarder travels a certain distance between  seconds and  seconds. If we define her velocity as , what is her distance in meters?

Possible Answers:

Correct answer:

Explanation:

We define velocity as the derivative of distance, or .

Thus, in order to find the distance traveled when given the velocity, we'll need to take the definite integral of the velocity function over a specified time period, or .

Since , we can use the Power Rule for Integrals

 for all ,

to find:

Example Question #21 : Applications In Physics

In 1D electromagnetism, , where  is voltage drop, and  is the electric field. and  and  are arbitrary bounds. 

In a capacitor,  is just a constant. Find the voltage drop for a constant electric field with strength  from 

Possible Answers:

Correct answer:

Explanation:

We can simply plug in the values into our first equation by:

Example Question #22 : Applications In Physics

In Electrical Engineering, the Laplace Transform is a heavily used integral transform. It is given by:

, where  is a number. 

Determine the Laplace Transform of 

Assume 

Possible Answers:

Correct answer:

Explanation:

 

By the fundamental theorem of calculus and since

 

Example Question #23 : Applications In Physics

In Electrical Engineering, the Laplace Transform is a heavily used integral transform. It is given by:

, where  is a number. 

Determine the Laplace transform of 

Assume 

Possible Answers:

Correct answer:

Explanation:

By the given formula:

By the fundamental theorem of calculus and because

                             

Example Question #24 : Applications In Physics

A particle's position is given by the following equation:

Here,  represents the particle's displacement from its starting point after  seconds. What is the particle's acceleration at ?

Possible Answers:

Correct answer:

Explanation:

The velocity of an object is defined as the derivative of its position with respect to time; in turn, the acceleration of an object is defined as the derivative of its velocity with respect to time. Since we've been given the equation defining the particle's position after  seconds, to determine its acceleration we must take the second derivative of this equation.

The first derivative with respect to time is

.

The second derivative with respect to time is

.

Now we simply substitute  for the new equation for the particle's acceleration to yield

Example Question #25 : Applications In Physics

Water flows into a certain pool at a rate of   for an hour, where  is measured in minutes. Find the amount of water that flows into the pool during the first  minutes.

Possible Answers:

Correct answer:

Explanation:

We can model the flow of water into the pool by a function of time; let  be the amount of water in the pool at time . Then  represents the rate at which water flows into the pool at time . By the net change theorem, evaluating the definite integral of  over an interval  yields the total change, or amount of, that quantity over that interval. Hence, we can determine how much water flowed into the pool over the first  minutes by evaluating the definite integral of  from  to , as follows:

Hence, the amount of water that flows into the pool during the first  minutes at this rate is .

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