Limits and Free Falling Objects

[pippintook]

Use the position function s(t)=-49t^(2) + 200 which gives the height of an object that has fallen from a height of 200 meters. The velocity at time t = a seconds is given by the limit as t goes to a = (s(a) - s(t))/(a-t).
At what velocity will the object impact the ground?


I used V^2 = 2gh and got 62.6 m/s, but am not sure if that is right or if there is another formula I should use.

 

[gabbagabbahey]

The question asks you to find $v(a)$ by computing the limit,

$$\lim_{t \to a} \frac{s(a)-s(t)}{a-t}$$

not by plugging it into a kinematics formula.

$v(a)$ is the instantaneous speed of the object at the time $t=a$. The V you just found is different; it is the speed of the object the instant before it hits the ground, not the speed at t=a. Try your hand at the limit above.

 

[Architeuthis Dux]

Your calculation using the formula V^2 = 2gh is correct. This formula is derived from the kinematic equation for velocity, v = u + at, where u is the initial velocity (in this case, 0 m/s), a is the acceleration due to gravity (9.8 m/s^2), and t is the time. By setting v = 0 (since the object will have a velocity of 0 at impact) and solving for t, we get t = sqrt(2h/g). Plugging in the given height of 200 meters and the acceleration due to gravity, we get t = sqrt(2*200/9.8) = 2.83 seconds.

To find the velocity at this time, we can use the position function s(t) = -49t^2 + 200. Plugging in t = 2.83 seconds, we get s(2.83) = -49(2.83)^2 + 200 = -400. From the limit given in the question, we can rewrite the velocity formula as v = (s(a) - s(t))/(a-t) = (-400 - 200)/(2.83 - a). As a approaches 2.83, the denominator approaches 0, so we can use the limit notation to find the velocity at this time: v = lim as t approaches 2.83 (-400 - 200)/(2.83 - a) = 62.6 m/s.

So, your calculation using the formula V^2 = 2gh was correct, and there is another way to find the velocity using the limit as well. Both methods give the same result, which is the velocity at which the object will impact the ground.