A person standing at the edge of a seaside cliff kicks a stone, horizontally over the edge with a velocity of 18 m/s. The cliff is 52 m above the waters surface
A. How long does it take for the stone to fall to the water
B. What is the vertical velocity component of the stone just before it hits the water
C. What is the horizontal velocity component of the stone just before it hits the water
D. What is the total velocity of the stone just before it hits the water ?
a. h = 0.5*g*T^2 = 52.
4.9*T^2 = 52,
T = 1.47 s.
b. V^2 = Vo^2 + 2g*d = 0 +29.6*52 = 1539,
V = 39. m/s.
c. Vx = 18 m/s.
d. Vt = sqrt(18^2 + 39^2) =
A. Hmm, let me calculate the time it takes for the stone to fall to the water. We can use the formula d = (1/2)gt^2 to find the time (t), where d is the distance (52 m) and g is the acceleration due to gravity (approximately 9.8 m/s^2). Let's see... *calculating sounds* ...Ah, the time it takes for the stone to fall is about 2.28 seconds.
B. Now, let's determine the vertical velocity component just before it hits the water. We know that the stone falls freely, so its vertical velocity component will be equal to the gravitational acceleration multiplied by the time it takes to fall. So, the vertical velocity component is approximately 22.55 m/s downward.
C. As the stone is kicked horizontally, its horizontal velocity component remains constant throughout its motion. Therefore, the horizontal velocity component just before it hits the water is the same as the initial horizontal velocity, which is 18 m/s.
D. To find the total velocity just before the stone hits the water, we can calculate the resultant velocity using the horizontal and vertical velocity components. We'll use the Pythagorean theorem, which states that the square of the resultant velocity is equal to the sum of the squares of the horizontal and vertical velocities. Hold on a second, let me crunch some numbers... *calculating sounds* ...Voila! The total velocity just before the stone hits the water is approximately 27.57 m/s.
A. To find the time it takes for the stone to fall to the water, we can use the equation:
s = ut + (1/2)gt^2
Where:
s = 52 m (height of the cliff above the water)
u = 0 m/s (initial vertical velocity of the stone)
g = 9.8 m/s^2 (acceleration due to gravity)
t = time (unknown)
Since the initial vertical velocity is 0 m/s, the first term in the equation becomes 0, simplifying the equation to:
s = (1/2)gt^2
Plugging in the values gives us:
52 = (1/2)(9.8)t^2
Rearranging the equation gives us:
t^2 = (52 * 2) / 9.8
t^2 = 104 / 9.8
t^2 = 10.612
Taking the square root of both sides gives us:
t ≈ 3.26 seconds
Therefore, it takes approximately 3.26 seconds for the stone to fall to the water.
B. The vertical velocity component just before the stone hits the water is equal to the final vertical velocity of the stone. We can use the equation:
v = u + gt
Where:
v = unknown (final vertical velocity of the stone)
u = 0 m/s (initial vertical velocity of the stone)
g = 9.8 m/s^2 (acceleration due to gravity)
t = 3.26 seconds (time taken to fall to the water)
Plugging in the values gives us:
v = 0 + (9.8)(3.26)
v ≈ 31.9488 m/s
Therefore, the vertical velocity component of the stone just before it hits the water is approximately 31.95 m/s (rounded to two decimal places).
C. The horizontal velocity of the stone remains constant throughout its motion, as there are no horizontal forces acting on it. Hence, the horizontal velocity component of the stone just before it hits the water is the same as the initial horizontal velocity.
Given that the stone was kicked horizontally with a velocity of 18 m/s, the horizontal velocity component just before it hits the water would also be 18 m/s.
D. The total velocity of the stone just before it hits the water is the combination of its horizontal and vertical velocities. We can use the Pythagorean theorem to calculate the magnitude of the total velocity:
v_total = √(v_vertical^2 + v_horizontal^2)
Where:
v_total = unknown (total velocity of the stone just before it hits the water)
v_vertical = 31.95 m/s (vertical velocity component)
v_horizontal = 18 m/s (horizontal velocity component)
Plugging in the values gives us:
v_total = √((31.95)^2 + (18)^2)
v_total = √(1017.6025 + 324)
v_total = √1341.6025
v_total ≈ 36.59 m/s (rounded to two decimal places)
Therefore, the total velocity of the stone just before it hits the water is approximately 36.59 m/s (rounded to two decimal places).
To answer these questions, we can break down the problem into two components: horizontal and vertical motion.
A. How long does it take for the stone to fall to the water?
To find the time, we need to consider the vertical motion of the stone. We know that the stone falls from a height of 52 m, and we can use the following equation of motion:
h = ut + (1/2)gt^2
where h is the height, u is the initial vertical velocity, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time.
Since the stone is kicked horizontally, its initial vertical velocity (u) is zero. The height (h) is 52 m, and g is 9.8 m/s^2.
Using the equation, we can rearrange it to solve for time (t):
h = (1/2)gt^2
52 = (1/2) * 9.8 * t^2
52 = 4.9 * t^2
t^2 = 52 / 4.9
t^2 ≈ 10.6122
t ≈ √10.6122
t ≈ 3.26 seconds
Therefore, it takes approximately 3.26 seconds for the stone to fall to the water.
B. What is the vertical velocity component of the stone just before it hits the water?
As the stone falls under the influence of gravity, the vertical component of its velocity increases linearly. Just before it hits the water, the vertical velocity will be equal to the final velocity.
We can use the equation:
v = u + gt
where v is the final vertical velocity, u is the initial vertical velocity (which is zero in this case), g is the acceleration due to gravity, and t is the time.
Substituting the known values, we have:
v = 0 + 9.8 * 3.26
v ≈ 31.9 m/s
Therefore, the vertical velocity component of the stone just before it hits the water is approximately 31.9 m/s.
C. What is the horizontal velocity component of the stone just before it hits the water?
Since the stone was kicked horizontally, its horizontal velocity (v) remains constant throughout its motion.
The horizontal velocity component of the stone just before it hits the water is the same as its initial horizontal velocity, which is given as 18 m/s.
Therefore, the horizontal velocity component of the stone just before it hits the water is 18 m/s.
D. What is the total velocity of the stone just before it hits the water?
The total velocity (V) at any point is given by the vector sum of its horizontal (Vx) and vertical (Vy) components. We can calculate the total velocity using the Pythagorean theorem:
V = √(Vx^2 + Vy^2)
Substituting the known values, we have:
V = √(18^2 + 31.9^2)
V ≈ √(324 + 1017.61)
V ≈ √1341.61
V ≈ 36.6 m/s
Therefore, the total velocity of the stone just before it hits the water is approximately 36.6 m/s.