# Particle Motion Problem: Finding Position and Velocity with Derivatives

• chocolatelover
In summary, the particle's position along the x axis is given by the equation x=1.5+2.50t-3.9t^2, with x in meters and t in seconds. To determine its position when it changes direction, take the derivative, set it equal to 0, and plug the resulting value of t back into the original equation. The position when it changes direction is 1.80 meters. To find the velocity when it returns to the position it had at t=0, use the equation vf^2=vi^2+2a(xf-xi) and plug in the values of vi=2.50, xi=0, and xf=1.9. The velocity
chocolatelover

## Homework Statement

A particle moves along the x axis. Its position is given by the equation x=1.5+2.50t-3.9t^2 with x in meters and t in seconds.
a. Determine its position when it changes direction
b. what is its velocity when it returns to the position it had at t=o

## The Attempt at a Solution

a. I would need to take the derivative and set it equal to 0, right?

f'(x)=2.50-7.8t=0
t=(-2.5/-7.8)

I then need to plug t back into the original equation, right?

Would the position when it changes direction be 1.80?

b. What is the velocity when it returns to the position it had at t=0?

2.50-7.8(0)=2.5

Thank you very much

Or would this work?

x=1.50-2.50t-3.90t^2

2.5-2(3.90)t

0=2.5-7.8t

t=.321

x=1.5+2.5(.321)-3.9(.321)^2
x=1.9m

vf^2=vi^2+2a(xf-xi)
=2.50^2+2(7.8)(0+1.9)
=6.0

Thank you very much

Yes, you are on the right track. To find the position when the particle changes direction, you would need to take the derivative of the position equation and set it equal to 0. This will give you the time at which the particle changes direction. Then, you can plug this time back into the original equation to find the position.

For part b, you are correct in finding the velocity when the particle returns to its initial position at t=0. This can be found by plugging in t=0 into the derivative equation. Keep in mind that the units for velocity will be in meters per second.

## 1. What is a particle motion problem?

A particle motion problem involves determining the position, velocity, and/or acceleration of an object (or particle) as it moves through space. It can also involve solving for other related quantities, such as time or displacement.

## 2. How can derivatives be used to solve particle motion problems?

Derivatives, specifically the first and second derivatives, can be used to find the velocity and acceleration of the particle, respectively. The position of the particle can then be found by integrating the velocity function, and the velocity function can be obtained by integrating the acceleration function.

## 3. What is the difference between position and velocity in a particle motion problem?

Position refers to the location of the particle at a specific point in time, while velocity refers to the rate at which the particle's position is changing at a given time. In other words, velocity is the derivative of position with respect to time.

## 4. What are some common applications of particle motion problems?

Particle motion problems are commonly used in physics and engineering to model the motion of objects, such as projectiles, vehicles, and celestial bodies. They can also be used in real-life scenarios, such as predicting the trajectory of a ball or determining the speed of a car.

## 5. What are some strategies for solving particle motion problems?

Some strategies for solving particle motion problems include drawing a diagram to visualize the problem, identifying known and unknown quantities, using the appropriate equations and formulas, and carefully setting up and solving the problem step by step. It can also be helpful to double check your calculations and units to ensure accuracy.

• Introductory Physics Homework Help
Replies
2
Views
374
• Introductory Physics Homework Help
Replies
11
Views
398
• Introductory Physics Homework Help
Replies
15
Views
1K
• Introductory Physics Homework Help
Replies
11
Views
1K
• Introductory Physics Homework Help
Replies
7
Views
1K
• Introductory Physics Homework Help
Replies
19
Views
1K
• Introductory Physics Homework Help
Replies
10
Views
1K
• Introductory Physics Homework Help
Replies
13
Views
987
• Introductory Physics Homework Help
Replies
5
Views
780
• Introductory Physics Homework Help
Replies
5
Views
254