Solve Few Simple Questions on Velocity and Differentiability

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In summary, velocity is a measure of an object's rate of change in position over time, calculated by dividing the change in position by the change in time. It is different from speed, which only measures an object's magnitude of motion. There is a difference between average velocity and instantaneous velocity, with average velocity being the average rate of change over a period of time and instantaneous velocity being the velocity at a specific moment in time. Differentiability is important in understanding velocity, as it allows us to find the instantaneous velocity at any point on a position-time graph. In real life, velocity is used to make predictions and solve problems related to the motion of objects.
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razored
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I have a few questions and I was wondering if someone could check my answers.


1) Given the velocity function http://texify.com/img/%5CLARGE%5C%21v%28t%29%20%3D-%28t%2B1%29%20%5Csin%7B%5Cfrac%7Bt%5E%7B2%7D%7D%7B2%7D%7D%20.gif ,[/URL] find all the times on hte open interval 0<t<3 where the particle changes direction. Justify your answer.

I said, the possible places where this occurs is when v=0. So I solved for v =0, and the only number that matches that domain is approximately 2.5 Are there any other answers I am missing? Is my idea of when the particle changes direction v=0 the entire reasoning to when the particle changes direction? I thought of an instance where v=0 but then particle doesn't change direction but rather keeps going in the same direction. Any help?

2)Suppose a function g is defined by http://texify.com/img/%5CLARGE%5C%21k%5Csqr%7Bx%2B1%7D%20%5C%20%5C%20%20for%20%5C%20%5C%200%5Cleq%20x%20%5Cleq%203%20%5C%5Cmx%2B2%20%5C%20%5C%20%20for%20%5C%20%5C%203%3C%20x%20%5Cleq%205%20.gif where k and m are constants. If g is differentiable at x=3, what are the values for k and m?

I was usually able to solve this but now there is one extra variable that I do not know how to get rid of. What do I do?
 
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  • #2
razored said:
1) Given the velocity function http://texify.com/img/%5CLARGE%5C%21v%28t%29%20%3D-%28t%2B1%29%20%5Csin%7B%5Cfrac%7Bt%5E%7B2%7D%7D%7B2%7D%7D%20.gif ,[/URL] find all the times on hte open interval 0<t<3 where the particle changes direction. Justify your answer.

I said, the possible places where this occurs is when v=0. So I solved for v =0, and the only number that matches that domain is approximately 2.5 Are there any other answers I am missing? Is my idea of when the particle changes direction v=0 the entire reasoning to when the particle changes direction? I thought of an instance where v=0 but then particle doesn't change direction but rather keeps going in the same direction. Any help?
I would also look at the case where v = 0 for I imagine the particle is moving forward when v > 0, moving backward when v < 0 and not moving when v = 0. Clearly v = 0 only when the sine term is 0. Can you give me an exact value for t between 0 and 3 that will make the sine term 0?

2)Suppose a function g is defined by http://texify.com/img/%5CLARGE%5C%21k%5Csqr%7Bx%2B1%7D%20%5C%20%5C%20%20for%20%5C%20%5C%200%5Cleq%20x%20%5Cleq%203%20%5C%5Cmx%2B2%20%5C%20%5C%20%20for%20%5C%20%5C%203%3C%20x%20%5Cleq%205%20.gif where k and m are constants. If g is differentiable at x=3, what are the values for k and m?

I was usually able to solve this but now there is one extra variable that I do not know how to get rid of. What do I do?
Remember that if g'(3) exists, then g is continuous at x = 3. With this additional piece of info., what values of k and m will make g continuous at x = 3?
 
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  • #3
1) [tex]\sqrt{2 * \pi }[/tex]
2)When k and m are equal?

Are those correct?
 
  • #4
The first one is correct. The second one is not: Just try it for m = k = 1. Here's a tip: Find the equation of the tangent of k * sqrt{x + 1} at x = 3 and try to manipulate it so that it takes the form mx + 2.
 

FAQ: Solve Few Simple Questions on Velocity and Differentiability

1. What is velocity and how is it calculated?

Velocity is a measure of the rate of change of an object's position over time. It is calculated by dividing the change in position by the change in time. The formula for velocity is v = (xf - xi) / (tf - ti), where v is velocity, xf is final position, xi is initial position, tf is final time, and ti is initial time.

2. What is the difference between average velocity and instantaneous velocity?

Average velocity is the average rate of change of an object's position over a period of time, while instantaneous velocity is the velocity of an object at a specific moment in time. Average velocity is calculated using the total distance and total time, while instantaneous velocity is calculated using the change in position and change in time at a specific point in time.

3. What is the relationship between velocity and speed?

Velocity and speed are related, but they are not the same thing. Velocity includes direction and is a vector quantity, while speed does not include direction and is a scalar quantity. In other words, velocity tells us both the speed and direction of an object's motion, while speed only tells us the magnitude of the object's motion.

4. How does differentiability relate to velocity?

Differentiability is a mathematical concept that describes the smoothness of a function. In the context of velocity, it refers to the ability to find the instantaneous velocity at any point on a position-time graph. A function is considered differentiable if it has a well-defined tangent at every point, which allows us to find the instantaneous velocity at that point.

5. How can I use velocity to solve problems in real life?

Velocity is a fundamental concept in physics and is used to describe the motion of objects in real life. It allows us to make predictions about the behavior of objects and can be used to solve problems such as calculating the distance traveled, determining the time it takes to reach a certain point, and predicting the future position of an object. In everyday life, velocity can be used to understand and analyze the movement of everything from cars on the road to airplanes in the sky.

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