https://file.aiquickdraw.com/mathbot/file/1729098983144a2i3lchj.jpg

Let's break down the problem step by step from the image you've uploaded.

---

### Part (a): Calculating the mass \( M \) of the object.

**Given Information:**

- The crash test car has a mass of \( 1000 \, \text{kg} \) and is moving at a speed of \( 12 \, \text{m/s} \).
- The object of unknown mass \( M \) is initially at rest.
- The collision is completely inelastic, meaning the two objects stick together after the collision.
- The velocity after the collision is given by the formula:

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

We apply the conservation of momentum for an inelastic collision:

\[
m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_{\text{final}}
\]

Substituting the values:

\[
(1000 \times 12) + (M \times 0) = (1000 + M) \times \frac{8}{1 + 5t}
\]

At \( t = 0 \), the velocity after the collision becomes:

\[
v_{\text{final}} = \frac{8}{1 + 5 \times 0} = 8 \, \text{m/s}
\]

Now, plugging this into the equation:

\[
12000 = (1000 + M) \times 8
\]

Solving for \( M \):

\[
12000 = 8000 + 8M
\]

\[
12000 - 8000 = 8M
\]

\[
4000 = 8M
\]

\[
M = 500 \, \text{kg}
\]

So, the mass of the object is \( M = 500 \, \text{kg} \).

---

### Part (b): Deriving the position \( x(t) \) of the car-object system.

We are given the velocity as a function of time:

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

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

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

Let’s solve the integral:

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

Given that the initial position \( x(0) = 0 \), we can find \( C \):

\[
x(0) = \frac{8}{5} \ln(1 + 5 \times 0) + C = 0
\]

\[
C = 0
\]

Thus, the position as a function of time is:

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

---

### Summary:

- The mass of the object is \( M = 500 \, \text{kg} \).
- The position of the car-object system as a function of time is \( x(t) = \frac{8}{5} \ln(1 + 5t) \).

Would you like further details or have any questions?

---

### Questions to explore:

1. How is momentum conserved in an inelastic collision?
2. What role does the velocity function \( v = \frac{8}{1 + 5t} \) play in determining the outcome of the collision?
3. How does the logarithmic position function affect the motion over time?
4. What are the assumptions made in using the conservation of momentum in this problem?
5. How would the result change if the collision was elastic instead of inelastic?

### Tip:
In inelastic collisions, kinetic energy is not conserved, but momentum is always conserved. This is key when solving collision problems.

A crash test car of mass 1000 kg moving at a constant speed of 12 m/s collides completely 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 by the expression v = 8 / (1 + 5t).

Explore a detailed AP Physics C problem involving a crash test car colliding inelastically with a stationary object. This problem focuses on conservation of momentum, velocity as a function of time, and the integration of velocity to determine position. Perfect for high school students studying momentum and collisions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation