https://file.aiquickdraw.com/mathbot/file/1729020265270fsjjiivc.jpg

Let's break down and solve each part of the problem step by step.

### Part (a): Calculate the mass \( M \) of the object.
We have a crash test car of mass \( 1000 \, \text{kg} \) moving at a speed of \( 12 \, \text{m/s} \). It collides inelastically with an object of unknown mass \( M \) that is initially at rest. After the collision, the speed \( v \) of the combined system is given by:

\[
v = \frac{8}{1 + 5t} \, \text{m/s}
\]

We know that in an inelastic collision, momentum is conserved. The initial momentum of the system is:

\[
\text{Initial momentum} = (m_1 \cdot v_1) + (m_2 \cdot v_2)
\]

where:
- \( m_1 = 1000 \, \text{kg} \) (mass of the car)
- \( v_1 = 12 \, \text{m/s} \) (velocity of the car)
- \( m_2 = M \) (mass of the object, unknown)
- \( v_2 = 0 \, \text{m/s} \) (velocity of the object, initially at rest)

Thus:

\[
\text{Initial momentum} = 1000 \cdot 12 + M \cdot 0 = 12000 \, \text{kg}\cdot\text{m/s}
\]

After the collision, the combined mass of the system is \( (1000 + M) \), and the velocity is \( v = \frac{8}{1 + 5t} \). Therefore, the final momentum is:

\[
\text{Final momentum} = (1000 + M) \cdot \frac{8}{1 + 5t}
\]

By conservation of momentum:

\[
12000 = (1000 + M) \cdot \frac{8}{1 + 5t}
\]

To find \( M \), rearrange and solve the equation:

\[
1000 + M = \frac{12000 \cdot (1 + 5t)}{8}
\]

\[
1000 + M = 1500 \cdot (1 + 5t)
\]

\[
M = 1500 \cdot (1 + 5t) - 1000
\]

This expression gives the mass \( M \) as a function of time \( t \).

### Part (b): Derive an expression for the position as a function of time.
The velocity of the system after the collision is given as:

\[
v(t) = \frac{8}{1 + 5t}
\]

To find the position \( x(t) \), we integrate the velocity function since:

\[
x(t) = \int v(t) \, dt
\]

\[
x(t) = \int \frac{8}{1 + 5t} \, dt
\]

Let \( u = 1 + 5t \). Then \( du = 5 \, dt \), and \( dt = \frac{du}{5} \):

\[
x(t) = \int \frac{8}{u} \cdot \frac{du}{5}
\]

\[
x(t) = \frac{8}{5} \int \frac{1}{u} \, du
\]

\[
x(t) = \frac{8}{5} \ln |u| + C
\]

Substituting back for \( u = 1 + 5t \):

\[
x(t) = \frac{8}{5} \ln |1 + 5t| + C
\]

Since the initial position \( x(0) = 0 \), we set \( C = 0 \):

\[
x(t) = \frac{8}{5} \ln(1 + 5t)
\]

### Summary of Answers:
1. **Mass of the object (M):**
   \[
   M = 1500 \cdot (1 + 5t) - 1000
   \]

2. **Position as a function of time (x(t)):**
   \[
   x(t) = \frac{8}{5} \ln(1 + 5t)
   \]

Would you like further details on any of these steps or have any other questions?

### Follow-up Questions:
1. How is momentum conserved in an inelastic collision, and how does it differ from an elastic collision?
2. What happens to the kinetic energy in an inelastic collision?
3. How can you verify the correctness of the derived position function using initial conditions?
4. What are the implications of the velocity function approaching zero as \( t \to \infty \)?
5. How would the solution change if the collision was elastic rather than inelastic?

### Tip:
Remember, when integrating a function, always consider the initial conditions to solve for any constants of integration accurately.

A crash test car of mass 1000 kg moving at constant speed of 12 m/s collides inelastically with an object of mass M at time t=0. The object was initially at rest. The speed v in m/s of the car-object system after the collision is given as a function of time t in seconds by the expression v = 8 / (1 + 5t). (a) Calculate the mass M of the object. (b) Assuming an initial position of x=0, derive an expression for the position of the car-object system after the collision as a function of time t.

This problem involves a car of mass 1000 kg colliding inelastically with another object and explores how to calculate the mass of the object using momentum conservation and how to derive the position of the system as a function of time using velocity integration.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation