A toy rocket is launched from a platform 2.1 meters above the ground in such a way this it’s height , h (in meters) after t seconds is given by the equation h=-4.9t^2+14.0t + 2.1. How long will it take for the rocket to hit the ground?
To find the time it takes for the rocket to hit the ground, we need to find the value of t when the height, h, is equal to zero.
The equation is: h = -4.9t^2 + 14.0t + 2.1
Setting h to zero: 0 = -4.9t^2 + 14.0t + 2.1
This equation is a quadratic equation, and we can solve it by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are: x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = -4.9, b = 14.0, and c = 2.1.
t = (-14.0 ± √(14.0^2 - 4(-4.9)(2.1))) / (2(-4.9))
Simplifying this equation, we get:
t = (-14.0 ± √(196.0 + 40.92)) / (-9.8)
t = (-14.0 ± √(236.92)) / (-9.8)
Now, we have to consider both the positive and negative solutions since time cannot be negative:
t = (-14.0 + √(236.92)) / (-9.8) or t = (-14.0 - √(236.92)) / (-9.8)
Simplifying these equations, we get:
t ≈ 4.13 seconds or t ≈ 0.312 seconds
Therefore, it will take approximately 4.13 seconds for the rocket to hit the ground.
To find out how long it takes for the rocket to hit the ground, we need to find the value of t when the height, h, is equal to zero.
Given the equation: h = -4.9t^2 + 14.0t + 2.1
Setting h to zero: 0 = -4.9t^2 + 14.0t + 2.1
Now we can solve this quadratic equation to find the values of t.
-4.9t^2 + 14.0t + 2.1 = 0
To factorize or use the quadratic formula, we need to make the coefficient of t^2 positive. We can do this by multiplying both sides of the equation by -1.
4.9t^2 - 14.0t - 2.1 = 0
Now, we can either factorize the equation or use the quadratic formula. Let's solve it using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 4.9, b = -14.0, and c = -2.1.
t = (14.0 ± √((-14.0)^2 - 4 * 4.9 * -2.1)) / (2 * 4.9)
Simplifying further:
t = (14.0 ± √(196 + 41.16)) / 9.8
t = (14.0 ± √(237.16)) / 9.8
t = (14.0 ± 15.4) / 9.8
Now we solve for the two possible values of t:
t1 = (14.0 + 15.4) / 9.8
t2 = (14.0 - 15.4) / 9.8
t1 = 29.4 / 9.8
t2 = -1.4 / 9.8
Simplifying further:
t1 = 3.00 seconds
t2 = -0.14 seconds
Since time cannot be negative, the only valid solution is t = 3.00 seconds.
Therefore, it will take 3.00 seconds for the rocket to hit the ground.
To find the time it takes for the rocket to hit the ground, we need to determine the value of t when the height, h, is equal to zero.
Given the equation for the height of the rocket, h = -4.9t^2 + 14.0t + 2.1, we can set it to zero and solve for t.
0 = -4.9t^2 + 14.0t + 2.1
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = -4.9, b = 14.0, and c = 2.1.
Plugging these values into the quadratic formula, we have:
t = (-(14.0) ± √((14.0)^2 - 4(-4.9)(2.1))) / (2(-4.9))
Simplifying further, we get:
t = (-14.0 ± √(196 - (-40.92))) / (-9.8)
t = (-14.0 ± √(236.92)) / (-9.8)
Now, we can calculate the values inside the square root:
t = (-14.0 ± √(236.92)) / (-9.8)
t = (-14.0 ± 15.39) / (-9.8)
This results in two possible values for t:
t1 = (-14.0 + 15.39) / (-9.8)
t2 = (-14.0 - 15.39) / (-9.8)
Calculating these values, we get:
t1 = 1.39 / (-9.8) ≈ -0.142
t2 = -29.39 / (-9.8) ≈ 2.997
Since time cannot be negative in this context, we can discard the negative value (-0.142). Therefore, the rocket will hit the ground approximately 2.997 seconds after it was launched.
So, it will take around 2.997 seconds for the rocket to hit the ground.