Question

A stone is thrown upward to just reach the top of a 50m office building.(a) calculate the initial speed that is required (b) how long does it take for the stone to return to the initial point of projection

Answer 1

[AI-generated answer — Bot GPT 3.5]

To solve this problem, we can use the equation of motion for vertical displacement:

h = u*t + (1/2)*g*t^2

where h is the vertical displacement, u is the initial velocity, t is the time, and g is the acceleration due to gravity.

(a) In order for the stone to reach the top of the building, its vertical displacement must be equal to the height of the building, which is 50m. So we have:

50m = u*t - (1/2)*g*t^2

At the maximum height, the stone's vertical displacement is zero. So setting h = 0, we have:

0 = u*t - (1/2)*g*t^2

Solving this quadratic equation for t:

(1/2)*g*t^2 - u*t = 0

t * (0.5g*t - u) = 0

Either t = 0 (which is not the time we're interested in) or:

0.5g*t - u = 0
u = 0.5g*t

We know that the acceleration due to gravity, g, is approximately 9.8 m/s^2. Substituting this value and t = time of flight/2 into the equation, we can calculate the initial speed required.

(b) To calculate the time it takes for the stone to return to the initial point of projection, we need to find the time of flight, which is double the time it takes to reach the maximum height. From the previous equation, we have:

t = 2*u/g

Substituting the value of u from equation (a):

t = 2*(0.5g*t)/g
t = 1s

Therefore, it would take 1 second for the stone to return to the initial point of projection.

Answer 2

[AI-generated answer — Step-by-Step Bot]

To solve this problem, we'll use the equations of motion. Firstly, let's break down the problem into two parts:

1. Calculating the initial speed required to reach the top of the building.
2. Calculating the time it takes for the stone to return to its initial point of projection.

Let's begin with part (a):

(a) Calculating the initial speed required:
We can use the following equation of motion to solve for the initial speed (u):
v² = u² + 2as

In this case, v (final velocity) is 0 m/s when the stone reaches the top of the building (as it momentarily stops before coming down).
a (acceleration) is -9.8 m/s² (taking downward direction as negative).
s (distance) is the height of the building, which is 50 m.

Plugging in these values into the equation, we get:
0² = u² + 2(-9.8)(50)

Simplifying the equation:
0 = u² - 980

Rearranging the equation:
u² = 980

Taking the square root of both sides:
u = √980

Using a calculator:
u ≈ 31.36 m/s (rounded to two decimal places)

Therefore, the initial speed required is approximately 31.36 m/s.

Moving on to part (b):

(b) Calculating the time it takes for the stone to return to the initial point of projection:
We can use the equation of motion to calculate the time (t) it takes for the stone to return to its initial point of projection:
v = u + at

In this case, v (final velocity) is 0 m/s when the stone reaches its initial point of projection.
a (acceleration) is -9.8 m/s² (taking downward direction as negative).
u (initial speed) is 31.36 m/s, as calculated in part (a).

Plugging in these values into the equation, we get:
0 = 31.36 + (-9.8)t

Simplifying the equation:
-31.36 = -9.8t

Dividing both sides of the equation by -9.8:
t = -31.36 / -9.8

Calculating t:
t ≈ 3.2 seconds (rounded to one decimal place)

Therefore, it takes approximately 3.2 seconds for the stone to return to its initial point of projection.

Answer 3

[AI-generated answer — Explain Bot]

To find the answers to your questions, we need to use some basic physics equations and apply the laws of motion. Let's break it down step by step:

(a) To calculate the initial speed (also known as the initial velocity) required for the stone to reach the top of a 50m building, we can use the kinematic equation for vertical motion:

v^2 = u^2 + 2as

Here:
v = final velocity (which is zero at the top of its trajectory since it momentarily stops)
u = initial velocity (what we're trying to find)
a = acceleration due to gravity (which is approximately -9.8 m/s^2)
s = vertical displacement (50m upwards in this case)

Plugging in the values, we get:

0 = u^2 + 2*(-9.8)*50

Simplifying the equation, we have:

0 = u^2 - 980

Rearranging the equation, we get:

u^2 = 980

Taking the square root of both sides, we find:

u = √980 m/s

Therefore, the initial speed required for the stone to reach the top of the building is approximately 31.3 m/s (rounded to one decimal place).

(b) To calculate how long it takes for the stone to return to the initial point of projection, we can use another kinematic equation:

v = u + at

Here:
v = final velocity (zero at the start and end points)
u = initial velocity (31.3 m/s in this case, upward)
a = acceleration due to gravity (-9.8 m/s^2, downward)
t = unknown (what we're trying to find)

Since the stone takes the same amount of time to reach the top and return, we can divide the total time by 2. So, the total time is t, and the time to reach the top is t/2.

Using the equation for the time to reach the top (t/2), we get:

0 = 31.3 + (-9.8) * (t/2)

Simplifying the equation, we have:

-31.3 = -4.9t

Dividing both sides by -4.9, we find:

t = 6.39 seconds (rounded to two decimal places)

Therefore, it takes approximately 6.39 seconds for the stone to return to its initial point of projection.