The question asks whether the average velocity of a ball dropped from the CN Tower (450 meters above the ground) over a time interval starting at $t = 0$ and ending at $t = 5$ seconds is given by the following limit expression:

$\lim_{h \to 0} \frac{4.9(5+h)^2 - 4.9(5)^2}{h}$

Problem Breakdown:

1. Distance Function: The distance fallen by the ball as a function of time $t$ is given by the equation $s = f(t) = 4.9t^2$. This comes from the equation of motion for free-fall under gravity (where $9.8 \, \text{m/s}^2$ is the acceleration due to gravity, and dividing by 2 gives $4.9$).

2. Average Velocity Over Time Interval: The average velocity over a time interval $[0, 5]$ is the change in distance divided by the change in time: $\text{Average velocity} = \frac{f(5) - f(0)}{5 - 0}$ where $f(t) = 4.9t^2$.

   Substituting into the above formula: $\text{Average velocity} = \frac{4.9(5)^2 - 4.9(0)^2}{5 - 0} = \frac{4.9(25)}{5} = 24.5 \, \text{m/s}$

3. Limit Expression: The limit provided in the problem, $\lim_{h \to 0} \frac{4.9(5+h)^2 - 4.9(5)^2}{h},$ is a form of the derivative of the function $f(t) = 4.9t^2$ at $t = 5$. This limit calculates the instantaneous velocity at $t = 5$, not the average velocity over the interval. Let’s compute the derivative.

4. Instantaneous Velocity: The derivative of $s = 4.9t^2$ with respect to $t$ gives the instantaneous velocity: $v(t) = \frac{d}{dt}[4.9t^2] = 9.8t.$ At $t = 5$, $v(5) = 9.8(5) = 49 \, \text{m/s}.$

Conclusion:

The limit provided in the question is not the average velocity but the instantaneous velocity at $t = 5$, which is 49 m/s. The correct average velocity over the interval $[0, 5]$ is 24.5 m/s.

Thus, the statement is false.

Would you like further details or have any questions?

Related Questions:

1. How is the equation $s = 4.9t^2$ derived for a free-falling object?
2. What is the difference between average velocity and instantaneous velocity?
3. How do you compute the derivative of a quadratic function?
4. What is the significance of the limit definition of a derivative?
5. How does gravity affect the velocity of an object in free fall?

Tip: The derivative provides the slope of the tangent line to the curve, which represents instantaneous velocity at a given time.