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a stone is thrown from the top of a tower which is 11m high and stands on horizontal ground the speed of projection is 12m/s at 60 degrees with the vertical downward find the time taken​

Sagot :

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):

  • t = ?

The kinematic equation that relates all four of these variables is:

  • Δx = v₀t + 1/2at²

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.