A person standing at the edge of a seaside
cliff kicks a stone over the edge with a speed
of 12 m/s. The cliff is 52 m above the water’s
surface, as shown.
12 m/s 52 m
How long does it take for the stone to fall
to the water? The acceleration of gravity is
9.81 m/s
2
.
Answer in units of s.
Well, this stone is about to have a pretty wild ride. It's like the stone is going on a permanent vacation from the cliffs to the water! Let's calculate how long this speedy stone takes to make the plunge.
To find the time it takes for the stone to fall, we can use the kinematic equation: Δy = vi*t + 0.5*a*t^2
Where:
Δy is the change in height (52 m in this case)
vi is the initial velocity (12 m/s downward)
a is the acceleration due to gravity (-9.81 m/s^2, since it's pulling the stone downwards)
t is the time it takes for the stone to hit the water (that's what we want to find out!)
Alright, let's plug in the values:
52 = (12 * t) + (0.5 * -9.81 * t^2)
Now we can simplify and solve for t:
52 = 12t - 4.905t^2
Rearranging the equation:
4.905t^2 - 12t + 52 = 0
Using the quadratic formula, I've got an answer for you:
t = (-(-12) ± √((-12)^2 - 4 * 4.905 * 52)) / (2 * 4.905)
After doing some calculations, the two possible solutions for t are approximately 0.78 seconds and 6.3 seconds.
Now, I should note that since we're dealing with time, we can't have a negative value. So the stone takes approximately 0.78 seconds to fall from the cliff to the water.
Remember, though, this is all assuming the stone didn't make any pit stops to enjoy the view or do some sightseeing along the way!
To find the time it takes for the stone to fall to the water, we can use the formula for the time it takes for an object to fall from a certain height:
t = sqrt((2 * h) / g)
Where:
h = height of the cliff = 52 m
g = acceleration due to gravity = 9.81 m/s^2
Plugging in the values, we get:
t = sqrt((2 * 52) / 9.81)
Simplifying this expression, we get:
t = sqrt(104 / 9.81)
t ≈ sqrt(10.6)
Using a calculator, we find that:
t ≈ 3.26 s
Therefore, it takes approximately 3.26 seconds for the stone to fall to the water.
To find the time it takes for the stone to fall to the water, we can use the kinematic equation:
s = ut + (1/2)gt^2
Where:
- s is the distance (52 m, because it represents the height of the cliff).
- u is the initial velocity (12 m/s, because that's the speed at which the stone is kicked).
- g is the acceleration due to gravity (-9.81 m/s^2, because it acts in the opposite direction to the stone's motion).
- t is the time we want to find.
Rearranging the equation, we have:
52 = (1/2)(-9.81)t^2 + 12t
Let's solve this quadratic equation for t:
Multiply everything by 2 to get rid of the fraction:
104 = -9.81t^2 + 24t
Rearrange the equation:
9.81t^2 - 24t + 104 = 0
Now we can solve this quadratic equation using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this equation:
- a = 9.81
- b = -24
- c = 104
Plugging in these values and simplifying, we get:
t = (-(-24) ± √((-24)^2 - 4(9.81)(104))) / (2(9.81))
t = (24 ± √(576 - 4074.24)) / 19.62
Calculating further, we have:
t = (24 ± √(-3498.24)) / 19.62
Since the discriminant is negative, the quadratic equation has no real solutions. This means the stone will never reach the water. Either there is some other factor affecting the motion (such as air resistance) or there may be an error in the given information.
Assuming a horizontal kick,
4.9 t^2 = 52
t = 3.26 s