Question

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​

Answer 1

t = 2.896 s

Step-by-step 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.

• 
  `\displaystyle t = (-b \pm √(b^2-4ac) )/(2a)`
• 
  `\displaystyle (-12* sin(60) \pm √([12* sin(60)]^2-4(-4.9)(11)) )/(2(-4.9))`
• 
  `\displaystyle (-12* sin(60) \pm √(323.6))/(-9.8)`

Split the equation into its positive and negative cases.

Positive:

• 
  `\displaystyle (-12* sin(60) + √(323.6))/(-9.8) = (7.596580615)/(-9.8) = -0.7751612872`

Negative:

• 
  `\displaystyle (-12* sin(60) - √(323.6))/(-9.8) = (-28.38119031)/(-9.8) = 2.896039828`

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.