Finding out maximum from given velocity?

In summary, the conversation discussed finding the time at which the kinetic energy is first at a maximum for a given velocity function. The individual was given the position and took the derivative to find the velocity function. They then graphed the velocity and kinetic energy and determined that the kinetic energy is proportional to v2. They also noted that the v2 values at the maxima of v are greater than the v2 values at the minima of v, and that a maximum in v occurs before a minimum in v after t = 0.
  • #1
Richard Ros
18
0

Homework Statement



Given that V(t) = π*cos(πt/1.6 + π/6), at which time after t = 0 s is the kinetic energy first at a maximum? The answer is 1.33 seconds. Although, when I put it in the calculator, 1.33s is at a minimum. How would you do this by hand? Step by step work would be appreciated. Thanks!

Homework Equations





The Attempt at a Solution



I was given the position which was x(t) = 1.6 sin(πt/1.6 + π/6) and took the derivative to get V(t) = π*cos(πt/1.6 + π/6). No clue on how to go from there to get to the maximum.
 
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  • #2
You derivative is not entirely correct. Check that.

But even what you have should be enough to solve the problem. Graph the velocity. You should get a sinusoid. Now graph the kinetic energy. What does it look like?
 
  • #3
I'm pretty sure my derivative is correct as there were more questions to this problem which they asked me the velocity at 0 seconds. As I plugged it in, I got the same answer as the answer key.
 
  • #4
Yes, it is correct indeed. Sorry about that. The rest of may message still holds.
 
  • #5
I graphed it but I really want to know how to mathematically determine the maximum or in this case the minimum. Which I think the answer key may of been wrong.
 

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  • #6
This is the graph of velocity. What is interesting, is the graph of kinetic energy.

What is the expression for the kinetic energy given the equation for velocity you have found?
 
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  • #7
The kinetic energy is proportional to v2. For your velocity variation, how do the v2 values at the maxima of v compare with the v2 values at the minima of v? Which occurs first after t = 0, a maximum in v or a minimum in v?
 

1. What is meant by "maximum" in terms of velocity?

The maximum velocity refers to the highest value or limit that can be achieved by an object in motion. It is the peak speed that an object can reach before its speed starts to decrease.

2. How is the maximum velocity determined or calculated?

The maximum velocity can be calculated by finding the first derivative of the displacement-time equation and setting it equal to zero. This will give the time at which the object reaches its maximum velocity. The value of the maximum velocity can then be found by plugging in this time into the velocity-time equation.

3. What factors affect the maximum velocity of an object?

The maximum velocity of an object can be affected by several factors, including the initial velocity, the acceleration, and the presence of external forces such as friction or air resistance. The mass and shape of the object can also play a role in determining its maximum velocity.

4. Can an object have a different maximum velocity in different directions?

Yes, an object can have a different maximum velocity in different directions. This is known as vector quantities. For example, a ball thrown upwards will have a maximum velocity in the upwards direction, but a different maximum velocity in the downwards direction when it falls back to the ground due to the effects of gravity.

5. How is the concept of maximum velocity applied in real-world situations?

The concept of maximum velocity is important in fields such as engineering, physics, and sports. It helps in designing machines and structures that can withstand high speeds, predicting the behavior of objects in motion, and improving athletic performance by finding the optimal speed for different activities.

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