Calculate Speed of Red Ball Falling From 26m

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A red ball is thrown downward from a height of 26 meters with an initial speed of 1.3 m/s, and the goal is to determine its speed just before it hits the ground. The equation used to find the time of impact is set to zero, but the initial attempt incorrectly calculated the time as 5.03 seconds instead of the correct 2.17 seconds. The error stemmed from misapplying the quadratic equation, particularly by incorrectly assuming t could equal zero. It was clarified that the quadratic formula should be used to find the correct roots. Correctly applying the quadratic formula will yield the accurate time and speed of the ball just before impact.
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Homework Statement


A red ball is thrown down with an initial speed of 1.3 m/s from a height of 26 meters above the ground. The force of gravity due to the Earth results in the balls each having a constant downward acceleration of 9.81 m/s2.

What is the speed of the red ball right before it hits the ground?

Homework Equations


x = 26 - 1.3t - 4.905t^2

The Attempt at a Solution


I get how to do this problem, but I wasn't able to get the correct answer the first time when I did it completely mathematically. Can someone check where I'm going wrong in this work?

We need to find the time that the ball hits the ground --> set x = 0
0 = 26 - 1.3t - 4.905t^2
-26 = -1.3t - 4.905t^2
-26 = (-1.3 - 4.905t)*t
t = 0,
-4.905t = -24.7
t = 5.03 seconds

I don't know where I'm going wrong to get that t = 5.03 seconds, when the answer is 2.17 seconds (by using a calculator and plotting the zeroes). Is it wrong to solve the quadratic like this when solving for the roots?
 
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Thewindyfan said:

Homework Statement


A red ball is thrown down with an initial speed of 1.3 m/s from a height of 26 meters above the ground. The force of gravity due to the Earth results in the balls each having a constant downward acceleration of 9.81 m/s2.

What is the speed of the red ball right before it hits the ground?

Homework Equations


x = 26 - 1.3t - 4.905t^2

The Attempt at a Solution


I get how to do this problem, but I wasn't able to get the correct answer the first time when I did it completely mathematically. Can someone check where I'm going wrong in this work?

We need to find the time that the ball hits the ground --> set x = 0
0 = 26 - 1.3t - 4.905t^2
-26 = -1.3t - 4.905t^2
-26 = (-1.3 - 4.905t)*t
t = 0,
-4.905t = -24.7
t = 5.03 seconds

I don't know where I'm going wrong to get that t = 5.03 seconds, when the answer is 2.17 seconds (by using a calculator and plotting the zeroes). Is it wrong to solve the quadratic like this when solving for the roots?

If ##-26 = t(-1.3-4.905 t)## you cannot have ##t = 0##. If you set ##t = 0## you get right-hand-side = 0 (because 0 times anything = 0), but left-hand-side = -26, and 26 ≠ 0.
So, yes, it is very wrong indeed to solve quadratic equations that way!

Why don't you just use the familiar quadratic equation roots formula?
 
Ray Vickson said:
If ##-26 = t(-1.3-4.905 t)## you cannot have ##t = 0##. If you set ##t = 0## you get right-hand-side = 0 (because 0 times anything = 0), but left-hand-side = -26, and 26 ≠ 0.
So, yes, it is very wrong indeed to solve quadratic equations that way!

Why don't you just use the familiar quadratic equation roots formula?
Yeah I guess I just confused that approach with solving the time for when x = some number. Thanks for pointing that out!
 
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