To find the velocity of a ball dropped from a tower using the definition of the derivative from first principles, we'll start by setting up the problem.
Let's denote the height of the ball at time t as h(t). In this case, the initial height is 40 m, and the time elapsed is 5 seconds. We want to find the velocity of the ball, which is the rate of change of height with respect to time.
The derivative of h(t) with respect to t, denoted as dh/dt or h'(t), represents the instantaneous velocity of the ball at time t.
Using the definition of the derivative from first principles, we have:
h'(t) = lim (Δt -> 0) [h(t + Δt) - h(t)] / Δt
To apply this definition, we need to find the height of the ball at time t + Δt and h(t). Let's substitute the given values:
h(t) = 40 m (initial height) t = 5 s (time elapsed)
Now, we'll choose a small value for Δt, let's say Δt = 0.001 s. With these values, we can calculate the height of the ball at t + Δt and h(t):
h(t + Δt) = height of the ball at t + Δt h(t) = 40 m
Now, we can substitute these values into the definition of the derivative:
h'(t) = lim (Δt -> 0) [h(t + Δt) - h(t)] / Δt = lim (Δt -> 0) [h(5 + 0.001) - 40] / 0.001
To complete the calculation, we need to know the mathematical expression for the height of the ball as a function of time, which will depend on the acceleration due to gravity.
If we assume that the ball is dropped in a vacuum (ignoring air resistance), the height of the ball at time t can be modeled using the equation:
h(t) = h(0) - (1/2) * g * t^2
Where: h(0) is the initial height (40 m), g is the acceleration due to gravity (approximately 9.8 m/s^2), t is the time elapsed.
With this equation, we can substitute the given values and evaluate the expression:
h'(5) = lim (Δt -> 0) [h(5 + 0.001) - 40] / 0.001
Using the height equation, h(t) = h(0) - (1/2) * g * t^2, we have:
h'(5) = lim (Δt -> 0) [(h(0) - (1/2) * g * (5 + 0.001)^2) - 40] / 0.001
Evaluating this limit will require further calculations and approximations, but this is the general process for finding the velocity of a ball dropped from a tower using the definition of the derivative from first principles in calculus.