A sphere is fired downwards into a medium with an initial speed of 27 m/s. If it experiences a deceleration of a = (-6t) m/s2, where ( is in seconds, determine the distance traveled before it stops.

To solve this problem, we need to determine the distance traveled by the sphere before it comes to a stop, given its initial speed and the deceleration it experiences.

### Step 1: Find the time when the sphere stops
The sphere stops when its velocity becomes zero. We can find the time at which this happens by integrating the acceleration function to get the velocity function.

Given:
- Initial velocity \( v_0 = 27 \, \text{m/s} \)
- Acceleration \( a(t) = -6t \, \text{m/s}^2 \)

The velocity \( v(t) \) is the integral of acceleration with respect to time:
\[
v(t) = \int a(t) \, dt = \int -6t \, dt = -3t^2 + C
\]
Where \( C \) is the constant of integration. To find \( C \), we use the initial condition:
\[
v(0) = v_0 = 27 \, \text{m/s}
\]
Substituting into the velocity equation:
\[
27 = -3(0)^2 + C \implies C = 27
\]
So the velocity function is:
\[
v(t) = -3t^2 + 27
\]

To find the time \( t \) when the sphere stops, set \( v(t) = 0 \):
\[
0 = -3t^2 + 27
\]
\[
3t^2 = 27 \implies t^2 = 9 \implies t = 3 \, \text{seconds}
\]

### Step 2: Calculate the distance traveled
The distance traveled is the integral of the velocity function over time from \( t = 0 \) to \( t = 3 \) seconds.

\[
\text{Distance} = \int_0^3 v(t) \, dt = \int_0^3 (-3t^2 + 27) \, dt
\]
\[
\text{Distance} = \left[ -t^3 + 27t \right]_0^3
\]
\[
\text{Distance} = \left( -(3)^3 + 27(3) \right) - \left( -(0)^3 + 27(0) \right)
\]
\[
\text{Distance} = \left( -27 + 81 \right) - 0 = 54 \, \text{meters}
\]

### Final Answer:
The distance traveled by the sphere before it stops is **54 meters**.

Would you like more details, or do you have any questions?

Here are 5 related questions you might be interested in:
1. How would the result change if the initial speed were different?
2. What if the deceleration were constant instead of varying with time?
3. How does the acceleration function affect the time it takes for the object to stop?
4. What would be the distance traveled if the object experienced a different form of deceleration, such as \(a = -6t^2\)?
5. How can we generalize this problem for any initial velocity and deceleration function?

**Tip:** When dealing with problems involving motion under variable acceleration, always integrate the acceleration to find the velocity and position functions.

Learn how to calculate the distance traveled by a sphere with an initial speed of 27 m/s, experiencing variable deceleration (-6t m/s^2). Follow a detailed solution involving kinematics and integration. Suitable for high school 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