Particle in motion

Question

A particle is in motion according to f(t)=9t/(t^2+9).

A) when is the particle at rest?

B) when is the particle moving in the positive direction?

C) when is the particle speeding up? Slowing down?

Thank you!

Answer 1

Hey Kayla!

For this problem our big idea is this:

Velocity is the derivative of Position.

Acceleration is the derivative of Velocity.

So given our function for the position of our particle: f(t) = 9t/(t²+9), we'll need to think about the following:

A) If the particle at rest, that means it's velocity is 0. So we can find f'(t) using the quotient rule. This will be a new function that tells us the particle's velocity at any time t. We want to know when the velocity is zero, so we can set it equal to zero and solve for t.

B.) If the particle is moving in the positive direction, it must have a positive value for velocity. The most efficient way to do this is with a "Sign Chart." Basically, use your answers from part A on a number line. Test values in between/around each of these values by plugging them into your velocity function. Any "window" where you get a positive value will be where your particle is moving in the positive direction.

C.) This one is probably the trickiest. A particle is "Speeding Up" when its velocity and acceleration have the same sign. It is "Slowing Down" when the velocity and accelerations have opposite signs. So you'll first have to find acceleration by taking the second derivative of our original function. Set it equal to zero and solve, and then make a new sign chart for acceleration.

Finally, you can compare your sign charts for B and C to find out where their signs are the same, and where they're different.

That was a lot! If anything was unclear, or you have any questions, feel free to shoot me a message!