A stone is thrown vertically upwards with a velocity of 14.7 meters per second from a window 29.4 meters above the ground. How long does it take to reach the ground?
THis is what I came up with :
Acceleration= -9.8
final velocity= 0
initial velocity = 14.7
therefore : a= (v-u)/t
your distance equation:
height = -4.9t^2 + 14t + 29.4
at ground, height = 0
-4.9t^2 + 14t + 29.4 = 0
divide by -7
.7t^2 - 2t - 4.2=0
times 10
7t^2 - 20t - 42 = 0
t = (20 ± √1576)/14
= appr 4.26 seconds (or a negative)
To find the time it takes for the stone to reach the ground, we can use the equation for vertical motion:
h = ut + (1/2)at^2
Where:
h = height above the ground
u = initial velocity
t = time
a = acceleration due to gravity
Given:
u = 14.7 m/s (upwards)
h = 29.4 m
a = -9.8 m/s^2 (acceleration due to gravity)
Since the stone is thrown upwards, the initial velocity is positive and the acceleration due to gravity is negative.
Let's substitute these values into the equation:
29.4 = 14.7t + (1/2)(-9.8)t^2
To solve this quadratic equation, we rearrange it by multiplying throughout by 2 to eliminate the fractions:
58.8 = 29.4t - 4.9t^2
Rearranging the equation, we get:
4.9t^2 - 29.4t + 58.8 = 0
Now we need to solve this quadratic equation. We can do this using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this equation:
a = 4.9
b = -29.4
c = 58.8
Substituting the values into the formula:
t = (-(-29.4) ± √((-29.4)^2 - 4(4.9)(58.8))) / (2(4.9))
Simplifying further:
t = (29.4 ± √(860.136)) / 9.8
Calculating the square root and solving for both cases, we get two possible values for time:
t1 = (29.4 + √(860.136)) / 9.8
t1 = 6.01 seconds (approximately)
t2 = (29.4 - √(860.136)) / 9.8
t2 = -0.936 seconds (approximately)
We discard the negative value because time cannot be negative in this context.
Therefore, it takes approximately 6.01 seconds for the stone to reach the ground.