The equation you provided is:

\[
\frac{d^2 s(t)}{dt^2} = -mg
\]

This is a second-order differential equation representing the acceleration of an object under constant gravitational force, where:
- \( s(t) \) is the position of the object as a function of time \( t \),
- \( g \) is the gravitational acceleration (approximately \( 9.8 \, \text{m/s}^2 \) on Earth),
- \( m \) is the mass of the object,
- \( \frac{d^2 s(t)}{dt^2} \) represents the second derivative of position with respect to time, which is the acceleration.

### Solving the Differential Equation:
Since the acceleration \( \frac{d^2 s(t)}{dt^2} \) is constant and equal to \( -mg \), we can integrate it twice to find the position \( s(t) \) as a function of time.

1. **First integration**: 
   \[
   \frac{ds(t)}{dt} = -mg t + C_1
   \]
   where \( C_1 \) is the constant of integration, which represents the initial velocity \( v_0 \) of the object at \( t = 0 \).

2. **Second integration**:
   \[
   s(t) = -\frac{mg}{2} t^2 + C_1 t + C_2
   \]
   where \( C_2 \) is another constant of integration, representing the initial position \( s_0 \) of the object at \( t = 0 \).

Thus, the general solution to the differential equation is:
\[
s(t) = -\frac{mg}{2} t^2 + v_0 t + s_0
\]

Let me know if you'd like further details on this process!

### Questions to Expand Understanding:
1. How would the solution change if air resistance was considered?
2. Can you explain the physical meaning of each constant in the solution?
3. What happens if the object is thrown upwards instead of being dropped?
4. How would you determine the initial velocity \( v_0 \) from real-world measurements?
5. How is this equation connected to the laws of motion?

**Tip:** Always keep track of the physical meaning behind the constants when solving differential equations to connect math with real-world scenarios.

Explore the solution to the second-order differential equation d^2 s(t) / dt^2 = -mg, which represents the motion of an object under constant gravitational force. This detailed explanation covers integration steps to find position, initial velocity, and initial position of the object. Suitable for undergraduate physics or math 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