A particle of mass m
, initially at rest at x=0
, is accelerated by a force that increases in time as F=Ct2
. Determine its velocity  v
  as a function of time.
Express your answer in terms of the given quantities.Determine its position  x
  as a function of time.
Express your answer in terms of the given quantities.

Let's break down the problem step by step.

### 1. Given information:
- The particle has mass \( m \).
- It starts at rest, so \( v(0) = 0 \) and \( x(0) = 0 \).
- The force acting on the particle increases with time as \( F(t) = Ct^2 \), where \( C \) is a constant.
- We need to determine the velocity \( v(t) \) and position \( x(t) \) as functions of time.

### 2. Newton's Second Law:

Newton's Second Law relates force to acceleration:
\[
F(t) = m a(t)
\]
where \( a(t) \) is the acceleration of the particle. We also know that \( a(t) = \frac{dv(t)}{dt} \).

Given that \( F(t) = Ct^2 \), we can substitute this into Newton's Second Law:
\[
m a(t) = Ct^2
\]
\[
m \frac{dv(t)}{dt} = Ct^2
\]
Now, solve for \( \frac{dv(t)}{dt} \):
\[
\frac{dv(t)}{dt} = \frac{C}{m} t^2
\]

### 3. Integrate to find \( v(t) \):

Integrating both sides with respect to time \( t \):
\[
v(t) = \int \frac{C}{m} t^2 \, dt
\]
\[
v(t) = \frac{C}{m} \int t^2 \, dt
\]
\[
v(t) = \frac{C}{m} \left( \frac{t^3}{3} \right) + v(0)
\]
Since the particle starts from rest, \( v(0) = 0 \). So:
\[
v(t) = \frac{C}{3m} t^3
\]

### 4. Determine the position \( x(t) \):

We know that velocity is the time derivative of position: \( v(t) = \frac{dx(t)}{dt} \). Therefore:
\[
\frac{dx(t)}{dt} = \frac{C}{3m} t^3
\]
Now, integrate this to find \( x(t) \):
\[
x(t) = \int \frac{C}{3m} t^3 \, dt
\]
\[
x(t) = \frac{C}{3m} \int t^3 \, dt
\]
\[
x(t) = \frac{C}{3m} \left( \frac{t^4}{4} \right) + x(0)
\]
Since \( x(0) = 0 \), the final expression for \( x(t) \) is:
\[
x(t) = \frac{C}{12m} t^4
\]

### Summary:

- The velocity of the particle as a function of time is:
\[
v(t) = \frac{C}{3m} t^3
\]

- The position of the particle as a function of time is:
\[
x(t) = \frac{C}{12m} t^4
\]

---

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How would the result change if the force was \( F = C t \) instead of \( F = C t^2 \)?
2. What is the acceleration \( a(t) \) as a function of time for this force?
3. How would you find the time when the particle reaches a specific velocity?
4. How would you calculate the work done by the force up to time \( t \)?
5. What if the particle had an initial velocity? How would that change the equations?

**Tip:** When dealing with time-dependent forces, always use Newton's Second Law and integrate step by step to find velocity and position.

A particle of mass m, initially at rest at x=0, is accelerated by a force that increases in time as F=Ct^2. Determine its velocity v as a function of time. Express your answer in terms of the given quantities. Determine its position x as a function of time. Express your answer in terms of the given quantities.

Learn how to determine the velocity and position of a particle with mass m, initially at rest, accelerated by a time-dependent force F=Ct^2. This step-by-step solution explains the application of Newton's Second Law, differential equations, and integration. Suitable for undergraduate physics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation