Searches related to a stone is projected vertically upward with a speed of 14m/s from a tower 100m high. find the maximum height attained and the speed with which it strikes the ground
How high does it go?
v = Vi -9.81 t
at top v = 0
0 = 14 -9.81 t
so
t = 14/9.81
then
h = 100 + 14 t - 4.9 t^2
now fall from h
v^2 = 2 g h
To find the maximum height attained and the speed with which the stone strikes the ground, we can use the equations of motion. Here are the steps to solve this problem:
Step 1: Determine the initial velocity:
The stone is projected vertically upward with a speed of 14 m/s. Therefore, the initial velocity (u) is +14 m/s (upward direction).
Step 2: Determine the final velocity at maximum height:
At the maximum height, the stone momentarily stops before falling down. Therefore, the final velocity (v) at maximum height is zero.
Step 3: Determine the acceleration:
The acceleration (a) due to gravity is constant and is approximately 9.8 m/s², acting in the downward direction.
Step 4: Calculate the time taken to reach the maximum height:
Using the equation of motion v = u + at, we can substitute the values v = 0 m/s, u = +14 m/s, and a = -9.8 m/s² (negative as it's acting downward). Solving for time (t):
0 = 14 - 9.8t
9.8t = 14
t = 14 / 9.8
t ≈ 1.43 seconds
Step 5: Calculate the maximum height attained:
Using the equation of motion s = ut + 0.5at², we can substitute the values s = ?, u = +14 m/s, a = -9.8 m/s², and t = 1.43 seconds. Solving for maximum height (s):
s = (14 * 1.43) + 0.5 * (-9.8) * (1.43)²
s ≈ 20.02 - 9.8 * 1.44
s ≈ 20.02 - 20.01
s ≈ 0.01 meters
Therefore, the maximum height attained by the stone is approximately 0.01 meters.
Step 6: Calculate the time taken to strike the ground:
When the stone strikes the ground, its height (s) will be 0 meters. Using the equation of motion s = ut + 0.5at², we can substitute the values s = 0, u = 14 m/s, and a = 9.8 m/s² (positive as it's acting downward). Solving for time (t):
0 = 14t + 0.5 * 9.8 * t²
0 = 14t + 4.9t²
0 = t(14 + 4.9t)
t = 0 (time at the start) or 14 + 4.9t = 0
4.9t = -14
t = -14 / 4.9 (ignoring negative time)
t ≈ -2.86 seconds
Step 7: Calculate the speed with which it strikes the ground:
Using the equation of motion v = u + at, we can substitute the values v = ?, u = 14 m/s, a = 9.8 m/s² (positive as it's acting downward), and t = -2.86 seconds. Solving for final velocity (v):
v = 14 + (9.8 * -2.86)
v ≈ 14 - 28.25
v ≈ -14.25 m/s
The negative sign indicates that the stone strikes the ground in the downward direction.
Therefore, the maximum height attained by the stone is approximately 0.01 meters, and the speed with which it strikes the ground is approximately 14.25 m/s in the downward direction.
To find the maximum height attained by the stone and the speed with which it strikes the ground, we can use the equations of motion.
Let's break down the problem into different parts and calculate step by step:
1. Find the time it takes for the stone to reach the maximum height:
We can use the equation:
v^2 = u^2 + 2as
where v is the final velocity (0 m/s at the highest point), u is the initial velocity (14 m/s), a is the acceleration (-9.8 m/s^2, considering gravity acting downwards), and s is the displacement (maximum height attained).
Rearranging the equation, we get:
0 = (14 m/s)^2 + 2(-9.8 m/s^2) * s
196 = 19.6s
s = 196 / 19.6
s = 10 m
Therefore, the maximum height attained by the stone is 10 meters.
2. Find the time taken to reach the maximum height:
We can use the equation:
v = u + at
where v is the final velocity (0 m/s at the highest point), u is the initial velocity (14 m/s), a is the acceleration (-9.8 m/s^2), and t is the time taken to reach the maximum height.
Substituting the values, we get:
0 = 14 m/s + (-9.8 m/s^2) * t
9.8 m/s^2 * t = 14 m/s
t = 14 m/s / 9.8 m/s^2
t = 1.43 seconds
Therefore, the time taken to reach the maximum height is approximately 1.43 seconds.
3. Find the time taken to reach the ground:
To find the time taken to reach the ground, we can use the equation:
s = ut + (1/2)at^2
where s is the displacement (100 m, as the stone fell from a height of 100 m), u is the initial velocity (0 m/s at the highest point), a is the acceleration (-9.8 m/s^2), and t is the time taken to reach the ground.
Substituting the values, we get:
100 m = 0 + (1/2) * (-9.8 m/s^2) * t^2
100 m = -4.9 m/s^2 * t^2
t^2 = -100 m / -4.9 m/s^2
t^2 ≈ 20.41
t ≈ √20.41
t ≈ 4.52 seconds
Therefore, the time taken to reach the ground is approximately 4.52 seconds.
4. Find the speed with which it strikes the ground:
We can use the equation:
v = u + at
where v is the final velocity (the speed at which it strikes the ground), u is the initial velocity (0 m/s at the highest point), a is the acceleration (-9.8 m/s^2), and t is the time taken to reach the ground.
Substituting the values, we get:
v = 0 + (-9.8 m/s^2) * 4.52 seconds
v ≈ -44.5 m/s
Since the negative sign indicates that the velocity is directed downwards, the speed, disregarding the direction, would be:
speed = |-44.5 m/s|
speed ≈ 44.5 m/s
Therefore, the speed with which the stone strikes the ground is approximately 44.5 m/s.