Velocity, Acceleration and Distance

In summary, the position of an object, x(t), is given by x(t) = A sin(ωt) where A is a constant and ω is the angular frequency. The instantaneous velocity, v(t), and acceleration, a(t), can be calculated by taking the derivatives of x(t). The expressions for v(t) and a(t) are v(t) = wAcos(wt) and a(t) = -w^2Asin(wt), respectively. To express v(t) and a(t) in terms of x, ω, and A, we can use the trigonometric identity sin^2(wt) + cos^2(wt) = 1 and solve for cos^2
  • #1
clipperdude21
49
0
1.11. If position of an object is given by: x(t) = A sin(ωt) where A is a constant and ω is
the angular frequency.
a) What is the instantaneous velocity at time t?
b) What is the instantaneous acceleration at time t?
c) Express the instantaneous velocity and the instantaneous acceleration in terms of
x, ω and A.




2. dx(t)/dt= v(t)
dv(t)/dt=a(t)




3. a) For (a) I just took the derivative of the x function and got wAcos(wt)
b) Same thing here but took the derivative for the answer i got in A and got
-w^2Asin(wt)
c) This is where i had some trouble. I got the instantaneous acceleration part of the
problem by substituting x for Asin(wt) and thus got a(t)=-w^2x. But i had no
clue how to get v(t) in terms of x,w,A.

What i did was set sin^2(wt) + cos^2(wt)=1 and solved for cos^2(wt). then plugged that into v^2(t)=w^2A^2cos^2(wt). I got v^2=w^2(A^2-x^2). After taking the square root i got v = +/-(w * sqrt(A^2-x^2). Is this right? should there be the +/- or is one ruled out?


Thanks for the help in advance!\sqrt{}
 
Last edited:
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  • #2
Good job! No, you can't pick one of the +/-. You need them both. At a given value of x, v can be either negative or positive depending on whether x is increasing or decreasing. And it could be doing either.
 
  • #3


I would like to provide an explanation and clarification for the given content. Velocity, acceleration, and distance are all important concepts in the study of mechanics and motion. Velocity refers to the rate of change of an object's position with respect to time. It is a vector quantity, meaning it has both magnitude and direction. Acceleration, on the other hand, is the rate of change of velocity with respect to time. It is also a vector quantity and is directly related to the net force acting on an object. Distance, also known as displacement, is the change in an object's position from its initial position to its final position.

In the given scenario, the position of an object is described by the function x(t) = A sin(ωt), where A is a constant and ω is the angular frequency. This function represents the object's position at any given time t. Now, let's answer the questions posed in the content.

a) The instantaneous velocity at time t can be found by taking the derivative of the position function with respect to time, which is v(t) = dx(t)/dt = Aωcos(ωt). This means that the instantaneous velocity at any given time t is equal to Aω times the cosine of ωt.

b) Similarly, the instantaneous acceleration at time t can be found by taking the derivative of the velocity function with respect to time, which is a(t) = dv(t)/dt = -Aω^2sin(ωt). This means that the instantaneous acceleration at any given time t is equal to -Aω^2 times the sine of ωt.

c) To express the instantaneous velocity and acceleration in terms of x, ω, and A, we can substitute the given position function into the expressions for velocity and acceleration. This gives us v(t) = Aωcos(ωt) = ω√(A^2-x^2) and a(t) = -Aω^2sin(ωt) = -ω^2x. The expression for velocity can be simplified using the Pythagorean identity, giving us v(t) = ω√(A^2-x^2). Thus, your solution for the velocity in terms of x, ω, and A is correct. There is no need for the +/- sign, as the velocity and acceleration are both negative when the object is moving in the opposite direction of the positive x
 

1. What is the difference between velocity and acceleration?

Velocity is a measure of an object's speed and direction, while acceleration is a measure of the change in an object's velocity over time. In other words, velocity tells us how fast an object is moving and in what direction, while acceleration tells us how much an object's velocity is changing.

2. How are velocity and acceleration related?

Velocity and acceleration are related in that acceleration is the rate of change of velocity. This means that if an object's velocity is changing, it is also experiencing acceleration. Additionally, if an object's acceleration is constant, its velocity will change at a constant rate over time.

3. How do you calculate velocity?

Velocity is calculated by dividing the distance an object travels by the time it takes to travel that distance. It is typically measured in units of distance per time, such as meters per second (m/s) or kilometers per hour (km/h).

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

Average velocity is calculated by dividing the total distance an object has traveled by the total time it took to travel that distance. It gives a general idea of how fast an object is moving over a certain period of time. Instantaneous velocity, on the other hand, is the velocity of an object at a specific moment in time. It is calculated by finding the slope of the tangent line to the object's position-time graph at that point.

5. How does distance relate to velocity and acceleration?

Distance is related to velocity and acceleration in that it is a measure of how far an object has traveled. Velocity and acceleration are both factors that can affect an object's distance traveled. For example, an object with a high velocity will cover more distance in a given amount of time than an object with a lower velocity. Similarly, an object with a high acceleration will cover more distance in a shorter amount of time than an object with a lower acceleration.

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