How Is Instantaneous Velocity Calculated in Particle Motion?

In summary: One caution: this particle does not have a minimum. s(0)= 5, the particle is at 5 meters after 0 seconds. s(3)= 0, the particle is at 0 meters after 3 seconds. s(6)= 5, the particle is at 5 meters after 6 seconds.
  • #1
Fusilli_Jerry89
159
0
A particle moves in a line so its position at any time in t ≥0 is given by the function s(t)=t²-6t+5,
where s is measured in meres and t is measured in seconds.

a) find the displacement during the first 6 seconds.
All i did was input 0 into the equation solve, then do the same with 6 and subtract the two. I got 0. Is this the right way to do that, os is that too "physics like"?

b)Find avg. velocity during first 6 seconds.
i basically did the same thing with (y2-y1)/(x2-x1) and got 0.

c) Find instantaneous velocity when t=4.
the derivative of the function is 2t-6 so:
s(4)=2(4)-6
=2

d) Find the acceleration of the particle when t=4
the derivative of 2t-6 is 2 so is it 2?

e) At what values does the particle change direction?
I got when t=3 but only by looking at the graph. How do you do this by calculus?

f) Where is the particle when s is a minimum?
again by using the graph
 
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  • #2
I moved this thread to the homework forums. Jerry, you've posted in the homework forums before, so you should know where to post. No warning points issued.

You also know you need to show your work. Please show your initial work on a-f and we'll try to help.

EDIT -- Oops, sorry, I missed the part where he showed some work on each...
 
Last edited:
  • #3
Looking at a graph that you've made is perfectly valid. Regardless, if you think about it, finding when the object changes direction will be at the max or minimum of the displacement function. To find a maximum or minimum you take the derivative and set it equal to zero, then solve for t.

For part e) you should probably say the value of time, and then the corresponding displacement, velocity, and acceleration at that time. Part f) is just making sure you realize you found a min in e).
 
  • #4
for -a-, use 6s in the equation...
 
  • #5
Fusilli_Jerry89 said:
A particle moves in a line so its position at any time in t ≥0 is given by the function s(t)=t²-6t+5,
where s is measured in meres and t is measured in seconds.

a) find the displacement during the first 6 seconds.
All i did was input 0 into the equation solve, then do the same with 6 and subtract the two. I got 0. Is this the right way to do that, os is that too "physics like"?
As long as you get the correct answer! No, evaluating a function is not too "physics like". Even mathematicians are allowed to do arithmetic!

b)Find avg. velocity during first 6 seconds.
i basically did the same thing with (y2-y1)/(x2-x1) and got 0.
Yes, the particle moved out to a distance -4 meters at t= 3 seconds and then came back. Since it was right back where it started, its average velocity was 0.


c) Find instantaneous velocity when t=4.
the derivative of the function is 2t-6 so:
s(4)=2(4)-6
=2
Good. The derivative is the instantaneous velocity here. Actually, if I were you teacher, I would deduct a point. The correct answer is " 2 meters per second". Do you see the difference?

d) Find the acceleration of the particle when t=4
the derivative of 2t-6 is 2 so is it 2?
Well, more precisely, the acceleration, when t= 4 is 2 meters per second per second.

e) At what values does the particle change direction?
I got when t=3 but only by looking at the graph. How do you do this by calculus?
You can't throw a car into reverse when you are doing 80 mph! The only way a particle can "turn around" is if its velocity changes sign and that means the velocity must be 0 at that point. You have already determined that the velocity at time t is 2t- 6. For what t is 2t- 6= 0?

f) Where is the particle when s is a minimum?
again by using the graph
If, at any t, the velocity is positive, the particle is still moving to the right and just a moment before, s was lower: not a minimum. If at any t, the velocity is negative, the particle is moving to the left and a moment later s will be lower: not a minimum. The velocity must be 0 at s minimum. You've already determined what t that is. What is s for that t?
 

Related to How Is Instantaneous Velocity Calculated in Particle Motion?

1. What is instantaneous velocity?

Instantaneous velocity is the rate at which an object is moving at a specific point in time. It is also known as the velocity at a specific moment or the velocity at a specific instant.

2. How is instantaneous velocity different from average velocity?

Instantaneous velocity is the velocity at a single point in time, while average velocity is the average rate of change in an object's position over a specific period of time. In other words, instantaneous velocity gives us the velocity at a specific moment, while average velocity gives us the overall picture of an object's motion.

3. How is instantaneous velocity calculated?

Instantaneous velocity can be calculated by finding the derivative of an object's position function with respect to time. This means finding the slope of the tangent line to the object's position-time graph at a specific point in time.

4. What is the importance of finding instantaneous velocity?

Instantaneous velocity is important because it allows us to understand an object's motion at a specific moment in time, rather than just its overall motion. This can be useful in analyzing the behavior of moving objects and predicting their future position and velocity.

5. How can I find the instantaneous velocity of an object in real life?

To find the instantaneous velocity of an object in real life, you can use various tools such as a stopwatch, a speedometer, or a motion sensor. By measuring the change in position of the object over a very small time interval, you can estimate its instantaneous velocity at that moment.

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