A stone is thrown at an angle of 30 degrees above the horizontal from the top of a cliff with an initial speed of 12 m/s. If it hits the ground 4.6 seconds later, what is the height of the cliff?
To find the height of the cliff, we need to analyze the motion of the stone and calculate the time it takes to reach the ground.
First, we can break down the initial velocity of the stone into its horizontal and vertical components.
The horizontal component of the velocity remains constant throughout the motion and is given by:
Vx = V × cos(θ)
where:
Vx is the horizontal component of velocity,
V is the initial speed of the stone,
and θ is the launch angle.
Substituting in the given values:
Vx = 12 m/s × cos(30°) = 10.392 m/s
The vertical component of the velocity changes due to the acceleration by gravity. We can calculate the time it takes to reach the highest point in the trajectory using the formula:
t = Vy / g
where:
t is the time of flight to reach the highest point,
Vy is the vertical component of velocity at launch,
and g is the acceleration due to gravity (approximately 9.8 m/s²).
Substituting in the given values:
t = 12 m/s × sin(30°) / 9.8 m/s² = 0.616 seconds
Since the total time of flight is 4.6 seconds, the time to reach the highest point must be half of this time:
t = 4.6 seconds / 2 = 2.3 seconds
Now we can determine the vertical component of the velocity at the highest point of the trajectory using the formula:
Vy = V × sin(θ) - g × t
Substituting in the given values:
Vy = 12 m/s × sin(30°) - 9.8 m/s² × 2.3 seconds
Vy = 6 m/s - 22.54 m/s = -16.54 m/s
The negative sign indicates that the velocity is directed downward at the highest point.
Next, we can calculate the height of the cliff using the formula:
h = Vy × t + 0.5 × g × t²
Substituting in the given values:
h = -16.54 m/s × 2.3 seconds + 0.5 × 9.8 m/s² × (2.3 seconds)²
h = -38.062 + 25.0194 = -13.0426
The negative sign indicates that the height is below the launching point.
However, we know that the height cannot be negative, so we take the absolute value:
h = | -13.0426 | = 13.0426 meters
Therefore, the height of the cliff is approximately 13.0426 meters.