Calculate the average velocity of the car for the time interval t = 0 to t = 10.0 s, given that the car's position is defined by x(t) = bt^2 - ct^3, where b = 3.00 m/s^2 and c = 0.... Calculate the average velocity of the car for the time interval t = 0 to t = 10.0 s, given that the car's position is defined by x(t) = bt^2 - ct^3, where b = 3.00 m/s^2 and c = 0.120 m/s^3.
Understand the Problem
The problem presents a scenario where a car accelerates from a stop. The car's position is given by the equation x(t) = bt^2 - ct^3, where b and c are constants. The question asks to calculate the average velocity of the car over the time interval from t = 0 to t = 10.0 seconds. To solve this, we need to find the displacement of the car during this time interval and divide it by the time elapsed.
Answer
$v_{av} = 18.0 \text{ m/s}$
Answer for screen readers
$v_{av} = 18.0 \text{ m/s}$
Steps to Solve
- Find the position at t = 0 s
First, we need to find the initial position of the car at $t = 0$.
$x(0) = b(0)^2 - c(0)^3 = 0$
- Find the position at t = 10.0 s
Next, we need to find the position of the car at $t = 10.0$ s. We are given $b = 3.00 m/s^2$ and $c = 0.120 m/s^3$.
$x(10) = b(10)^2 - c(10)^3 = 3.00(10)^2 - 0.120(10)^3 = 3.00(100) - 0.120(1000) = 300 - 120 = 180 \text{ m}$
- Calculate the displacement
The displacement $\Delta x$ is the difference between the final and initial positions:
$\Delta x = x(10) - x(0) = 180 - 0 = 180 \text{ m}$
- Calculate the average velocity
The average velocity $v_{av}$ is the displacement divided by the time interval $\Delta t = 10.0 - 0 = 10.0$ s.
$v_{av} = \frac{\Delta x}{\Delta t} = \frac{180}{10} = 18.0 \text{ m/s}$
$v_{av} = 18.0 \text{ m/s}$
More Information
The average velocity represents the constant velocity that the car would need to travel at to cover the same displacement in the same amount of time.
Tips
A common mistake would be to find the instantaneous velocity at t=10s, but that is NOT required, and is a different question from this one. Avoid calculating the derivative, and remember to simply calculate $x(t)$ at t=0 and t=10.
AI-generated content may contain errors. Please verify critical information
