A stone is thrown up with a velocity of 20 m/s. While coming down, it strikes a glass pan, held at half the height through which it has risen and loses half of its velocity in breaking the glass.
Find the velocity with which it will strike the ground.
A stone is thrown up with a velocity of 20 m/s from the ground. While coming down, it strikes a glass pan, held at half the height through which it has risen and loses half of its velocity in breaking the glass. Find the velocity with which it will strike the ground.
Could anyone solve it???
14.14m/s
Well, this stone seems like quite the troublemaker! Let's see if we can figure this out with a pinch of humor.
First, let's break down the situation like the glass pan (pun intended). The stone is thrown up with a velocity of 20 m/s. Now, while coming down, it decides to break the glass pan like a true drama queen and loses half of its velocity.
If we take the velocity at the maximum height to be 'v' (when it hits the glass pan), then the velocity after breaking the glass pan will be 'v/2'. Now, because the glass pan is held at half the height the stone has risen, the stone has fallen half of the total distance it went up.
Now, you may think this is a trap and I'm going to make a joke about gravity, but I won't fall for that one! Instead, let's focus on the fact that when the stone hits the ground, its total displacement will be the same as when it was thrown up.
Using a hefty dose of physics equations, we can apply the conservation of energy here. At the maximum height, the stone has potential energy equal to its kinetic energy. So, we have:
mgh = (1/2)mv^2
Where 'm' is the mass of the stone, 'g' is the acceleration due to gravity, 'h' is the maximum height, and 'v' is the velocity with which it hits the ground.
Now, if we put the new velocity after breaking the glass pan as 'v/2' and the total distance traveled as '2h', we'll have:
mgh = (1/2)mv^2/4
If we simplify and solve for 'v', we get:
v = 2√(gh)
So, with some math and humor combined, the velocity with which the stone will strike the ground is equal to 2 times the square root of the product of 'g' (the acceleration due to gravity) and 'h' (the maximum height).
Remember, my friend, don't let the stone trick you next time!
To find the velocity with which the stone will strike the ground, we can break down the problem into two parts: the first part is when the stone is thrown up, and the second part is when the stone is coming down after hitting the glass pane.
Let's determine the maximum height that the stone reaches. We can use the formula for the maximum height of a projectile:
h = (v^2 * sin^2(theta)) / (2 * g)
Since the stone is thrown up vertically, the angle of projection (theta) is 90 degrees, and the sine of 90 degrees is equal to 1. Also, the vertical component of initial velocity (v) is given as 20 m/s. The acceleration due to gravity (g) is approximately 9.8 m/s^2.
Therefore, substituting the values into the formula, we have:
h = (20^2 * 1^2) / (2 * 9.8)
h = 200 / 19.6
h ≈ 10.204 m
Now, we need to find the time it takes for the stone to reach the maximum height. We can use the following formula:
t = v * sin(theta) / g
Since the stone is thrown up vertically, the angle of projection (theta) is 90 degrees, and the sine of 90 degrees is equal to 1. We already know the vertical component of initial velocity (v) is 20 m/s, and the acceleration due to gravity (g) is approximately 9.8 m/s^2.
Therefore, substituting the values into the formula, we have:
t = 20 * 1 / 9.8
t ≈ 2.041 seconds
Now, we can find the time taken for the stone to reach halfway down after hitting the glass pane. Since the initial velocity after hitting the glass pane is half of its original velocity, the new initial velocity (v') is 20 / 2 = 10 m/s.
Using the formula for time of flight of a projectile:
T = 2 * t
Substituting the value of t, we have:
T = 2 * 2.041
T ≈ 4.082 seconds
Finally, we can find the final velocity (v') with which the stone hits the ground using the formula for final velocity of a projectile:
v' = v + g * T
Substituting the values of v, g, and T, we have:
v' = 10 + 9.8 * 4.082
v' ≈ 51.604 m/s
Therefore, the stone will strike the ground with a velocity of approximately 51.604 m/s.