Answer:
t = 2.896 s
Explanation:
Assuming the positive direction is upwards and the negative direction is downwards:
The stone has a displacement of -11 m after landing on the ground. The stone starts with an initial velocity of 12 m/s at a 60-degree angle, which we will need to break into its y-component (multiply by sine of the angle).
Assuming that air resistance is negligible, we can say that the stone is in free-fall, and therefore, the acceleration is the pull due to gravity (g = 9.8 m/s²). The acceleration is always acting in the downwards direction when the object is in projectile/free-fall motion (it is negative in this case).
We have three known variables:
- v₀ = 12 * sin(60) m/s
- Δx = -11 m
- a = -9.8 m/s²
We want to solve for the fourth variable (time):
The kinematic equation that relates all four of these variables is:
Substitute the known variables into the equation and solve for time.
- -11 = [12 * sin(60)] t + 1/2(-9.8)t²
- -11 = [12 * sin(60)] t - 4.9t²
- 0 = -4.9t² + [12 * sin(60)] t + 11
Use the quadratic formula to solve for t.
- [tex]\displaystyle t = \frac{-b \pm \sqrt{b^2-4ac} }{2a}[/tex]
- [tex]\displaystyle \frac{-12\times sin(60) \pm \sqrt{[12\times sin(60)]^2-4(-4.9)(11)} }{2(-4.9)}[/tex]
- [tex]\displaystyle \frac{-12\times sin(60) \pm \sqrt{323.6}}{-9.8}[/tex]
Split the equation into its positive and negative cases.
Positive:
- [tex]\displaystyle \frac{-12\times sin(60) + \sqrt{323.6}}{-9.8} = \frac{7.596580615}{-9.8} = -0.7751612872[/tex]
Negative:
- [tex]\displaystyle \frac{-12\times sin(60) - \sqrt{323.6}}{-9.8} = \frac{-28.38119031}{-9.8} = 2.896039828[/tex]
Time can never be negative, so we know the correct time is t = 2.896.
The stone takes 2.896 seconds to reach the ground.
