Particle's Velocity in t: -kt^3 + c

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In summary, the particle in question has a velocity given by v=-k/2t^2 + 5, where k is a constant that can be solved for using the given values of velocity and time. The particle has an acceleration that is inversely proportional to t^3, with a limiting velocity of 5ms^-1.
  • #1
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Homework Statement


A particle moves in a straight line with acceleration which is inversely proportional to t3 , where t is the time. The particle has a velocity of 3ms-1 when t=1 and its velocity approaches a limiting value of 5ms-1 . Find an expression for its velocity at time t.

Homework Equations


a=-kt^3

The Attempt at a Solution


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*****ROOKIE MISTAKE SHOULD BE -2/t^2 + d (last equation)***
 
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  • #2
I won't answer the question directly for you, but this is just a basic algebra problem, remember that when y is proportional to x, you use the formula y=kx where k is some constant, inversly proportional to x and you have y = k/x. Once you know how y is proportional to x, and you have test values for y and x, you can plug them in and solve for k.
 
  • #3
agent_509 said:
I won't answer the question directly for you, but this is just a basic algebra problem, remember that when y is proportional to x, you use the formula y=kx where k is some constant, inversly proportional to x and you have y = k/x. Once you know how y is proportional to x, and you have test values for y and x, you can plug them in and solve for k.

I did some more and got stuck again, do you mind looking at it?
 
  • #4
I'm not entirely sure what you did there here's what would be a good idea:

∫ dv/dt dt = k∫1/t^2 dt

v= -k/2t^2 + d

now you know that when t=1 v=3, and you also know that the limiting velocity (when t→∞) is 5, so when t→∞, d=v, and v= 5, so d = 5, so now you have the formula

v=-k/2t^2 + 5, and you know that when t=1, v=3.

so just plug it in and solve for k
 
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  • #5

I would first analyze the given information and equations to understand the behavior of the particle. The given acceleration equation, a=-kt^3, indicates that the acceleration of the particle decreases as time increases (since t^3 increases as t increases). This is consistent with the fact that the particle's velocity approaches a limiting value of 5ms-1 as time goes on.

Using the given information that the particle's velocity is 3ms-1 at t=1, we can plug this into the velocity equation, v=-kt^3 + c, to solve for the constant c. This gives us c=3+k.

Now, to find the expression for velocity at any time t, we can plug in the value of c into the velocity equation and solve for v. This gives us v=-kt^3 + 3+k.

To find the value of k, we can use the fact that the velocity approaches a limiting value of 5ms-1 as t goes to infinity. This means that when t is a very large number, the term -kt^3 becomes negligible and we are left with v=5. Plugging this into the velocity equation, we get 5=3+k, which gives us k=2.

Therefore, the final expression for the particle's velocity at time t is v=-2t^3 + 3+2 = -2t^3 + 5 ms-1. This equation shows that the velocity of the particle decreases as t increases, and approaches a limiting value of 5ms-1 as t goes to infinity.
 

1. What is the meaning of the constants k and c in the equation -kt^3 + c?

The constant k represents the acceleration of the particle, while c represents its initial velocity at time t=0.

2. How does the velocity of the particle change over time in the equation -kt^3 + c?

The velocity of the particle decreases over time, as evidenced by the negative coefficient of t^3. This means that the particle is slowing down as time goes on.

3. Can the velocity of the particle ever be positive in the equation -kt^3 + c?

No, the velocity of the particle will always be negative in this equation. This indicates that the particle is moving in the opposite direction of the initial velocity, slowing down over time.

4. What happens to the particle's velocity as time approaches infinity in the equation -kt^3 + c?

As time approaches infinity, the velocity of the particle will approach -infinity. This means that the particle will continue to slow down without ever coming to a complete stop.

5. How does the value of k affect the particle's velocity in the equation -kt^3 + c?

The value of k affects the rate at which the particle's velocity decreases over time. A larger value of k will result in a faster decrease in velocity, while a smaller value of k will result in a slower decrease in velocity.

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