How to find the maximum velocity and maximum acceleration?

In summary, the position of a particle moving along an x-axis is given by x=12t^2-2t^3. To find the maximum positive coordinate, we can take the derivative of the position function and solve for t. To find the maximum velocity and acceleration, we can repeat the same process with the velocity and acceleration functions, respectively. The fourth derivative of time is known as jerk.
  • #1
dumpman
17
0

Homework Statement


The position of a particle moving along an x-axis is given by x=12t^2-2t^3, what is the maximum positive coordinate, velocity and acceleration reached by the particle?

Homework Equations





The Attempt at a Solution


I took derivative of the quadratic and got V=24t-6t^2 and set it =0, then i solve for t and substituted in x=12t^2-2t^3 to get the max. positive coordinate.
But I don't know what to do to solve for the max velocity and acceleration:confused:
 
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  • #2
You have the velocity as a function of time, just like you had the position as a function of time.
 
  • #3
more hints?
 
  • #4
Why not repeat the same process for your function of velocity?
 
  • #5
Oh, so i set a=0 to find max velocity, but then what do I do to find the max acceleration?
 
  • #6
Find the derivative of acceleration as a function of time and set it to zero. The physical term for the fourth derivative time is called jerk.
 

1. How do you calculate the maximum velocity?

The maximum velocity can be calculated by taking the derivative of the position function with respect to time and setting it equal to 0. This will give you the critical points, which can then be plugged into the second derivative to determine if it is a maximum or minimum.

2. What factors affect the maximum velocity?

The maximum velocity can be affected by several factors such as the initial velocity, the acceleration, and the direction of the velocity. Other factors may include external forces, friction, and air resistance.

3. How can I graph the maximum velocity and acceleration?

To graph the maximum velocity and acceleration, you can plot the position, velocity, and acceleration functions on the same graph. The maximum velocity will be at the point where the velocity function reaches its peak, and the maximum acceleration will be at the point where the acceleration function reaches its peak.

4. Can the maximum velocity and acceleration be different for the same object?

Yes, the maximum velocity and acceleration can be different for the same object. This is because the maximum velocity is determined by the rate of change of the position, while the maximum acceleration is determined by the rate of change of the velocity. Therefore, the maximum velocity may occur at a different time than the maximum acceleration.

5. How can I use the maximum velocity and acceleration in real-world scenarios?

The maximum velocity and acceleration can be used in various real-world scenarios, such as designing roller coasters, analyzing car accidents, and determining the maximum speed of a projectile. They can also be used in sports, such as calculating the maximum speed of a baseball pitch or a sprinter's acceleration off the starting block.

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