Question

An object falls from rest with an acceleration of g.The uncertainty in the value of the time is 6℅ and the uncertainty in the value of g is 4℅ the best estimate for the uncertainty of the position of the object is

Answer 1

Let the position of object is h
time taken to reach the ground is t
object falls from height h ,
initial velocity , u = 0
use formula , S = ut + 1/2 at²
here, S = -h , u = 0, a = -g
then, -h = 0 - 1/2 ×gt²
h = 1/2 gt²

hence, expression is h = 1/2 gt²
so, to find error ,
% error in h = % error in g + 2 × % error in t

given, % error in g = 4 %
% error in time , t = 6 %

so, % error in h = 4 + 2 × 6 = 4 + 12 = 16%

hence, uncertainty of the position of the object is 16%

Answer 2

Given:

Acceleration = g

The uncertainty in the value of the time = 6℅

The uncertainty in the value of gravity = 4℅

To find:

The best estimate for the uncertainty of the position of the object

Solution:

Consider,

The object falls from a height h for the time t and reaches the ground.

According to the laws of motion,

Initial velocity = 0

Distance travelled = Initial velocity * Time + 1/2 * Accelration due to gravity * Time taken^2

Here,

Distance traveled = 0

Initial velocity = 0

Gravity is negative

Then,

Position = 1/2 * Accelration due to gravity * Time taken^2

To find the error,

% Error in position = % Error in gravity + 2 * % error in Time taken

4 + 2 × 6

Uncertainty in position= 16 %

Therefore, the uncertainty of the position of the object is 16 %.