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Answer :
To solve the problem of finding how many seconds after launch the toy rocket will reach the ground, let's start by analyzing the initial conditions:
1. Initial Velocity: The rocket is launched with an initial velocity of 60 ft/s.
2. Initial Height: The rocket is launched from a height of 3 ft above the ground.
3. Acceleration due to Gravity: The rocket experiences a downward acceleration of -16 ft/s².
We need to determine when the height of the rocket becomes zero, which means it has reached the ground.
### Step-by-step Solution:
To find the time it takes for the rocket to hit the ground, we can use the following kinematic equation for the position of an object under constant acceleration:
[tex]\[ \text{height} = \text{initial height} + (\text{initial velocity} \times t) + \frac{1}{2} \times \text{acceleration} \times t^2 \][/tex]
We set the height to zero (since we're interested in when it hits the ground) and substitute the known values:
[tex]\[ 0 = 3 + 60t - 8t^2 \][/tex]
This simplifies to the quadratic equation:
[tex]\[ -8t^2 + 60t + 3 = 0 \][/tex]
To solve this quadratic equation, we apply the quadratic formula:
[tex]\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Where [tex]\( a = -8 \)[/tex], [tex]\( b = 60 \)[/tex], and [tex]\( c = 3 \)[/tex]. Plug these values into the formula:
1. Calculate the discriminant: [tex]\( b^2 - 4ac = 60^2 - 4(-8)(3) = 3600 + 96 = 3696 \)[/tex].
2. Find the square root of the discriminant: [tex]\( \sqrt{3696} \)[/tex].
3. Calculate the two possible solutions for [tex]\( t \)[/tex]:
- [tex]\( t_1 = \frac{-60 + \sqrt{3696}}{-16} \)[/tex]
- [tex]\( t_2 = \frac{-60 - \sqrt{3696}}{-16} \)[/tex]
Since time cannot be negative, we choose the positive time value. The calculations show that the valid time for when the rocket hits the ground is approximately 7.55 seconds.
Therefore, the correct answer based on our calculations (and some rounding) matches one of the provided options, which is approximately 7.55 seconds. However, since this is not one of the answer choices given, it appears there might be a slight discrepancy.
Given the answer choices and our approximation:
- 0.05 s
- 2.03 s
- 3.80 s
- 3.70 s
The correct choice, considering our numerical result, should indeed be another number close to 7.55 seconds, which was computed in our solution steps.
1. Initial Velocity: The rocket is launched with an initial velocity of 60 ft/s.
2. Initial Height: The rocket is launched from a height of 3 ft above the ground.
3. Acceleration due to Gravity: The rocket experiences a downward acceleration of -16 ft/s².
We need to determine when the height of the rocket becomes zero, which means it has reached the ground.
### Step-by-step Solution:
To find the time it takes for the rocket to hit the ground, we can use the following kinematic equation for the position of an object under constant acceleration:
[tex]\[ \text{height} = \text{initial height} + (\text{initial velocity} \times t) + \frac{1}{2} \times \text{acceleration} \times t^2 \][/tex]
We set the height to zero (since we're interested in when it hits the ground) and substitute the known values:
[tex]\[ 0 = 3 + 60t - 8t^2 \][/tex]
This simplifies to the quadratic equation:
[tex]\[ -8t^2 + 60t + 3 = 0 \][/tex]
To solve this quadratic equation, we apply the quadratic formula:
[tex]\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Where [tex]\( a = -8 \)[/tex], [tex]\( b = 60 \)[/tex], and [tex]\( c = 3 \)[/tex]. Plug these values into the formula:
1. Calculate the discriminant: [tex]\( b^2 - 4ac = 60^2 - 4(-8)(3) = 3600 + 96 = 3696 \)[/tex].
2. Find the square root of the discriminant: [tex]\( \sqrt{3696} \)[/tex].
3. Calculate the two possible solutions for [tex]\( t \)[/tex]:
- [tex]\( t_1 = \frac{-60 + \sqrt{3696}}{-16} \)[/tex]
- [tex]\( t_2 = \frac{-60 - \sqrt{3696}}{-16} \)[/tex]
Since time cannot be negative, we choose the positive time value. The calculations show that the valid time for when the rocket hits the ground is approximately 7.55 seconds.
Therefore, the correct answer based on our calculations (and some rounding) matches one of the provided options, which is approximately 7.55 seconds. However, since this is not one of the answer choices given, it appears there might be a slight discrepancy.
Given the answer choices and our approximation:
- 0.05 s
- 2.03 s
- 3.80 s
- 3.70 s
The correct choice, considering our numerical result, should indeed be another number close to 7.55 seconds, which was computed in our solution steps.
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