Nicole throws a ball straight up. Chad watches the ball from a window 5.0m above the point where Nicole released it. The ball passes Chad on the way up, and it has a speed of 10m/s as it passes him on the way back down. How fast did Nicole throw the ball?
Well, it sounds like Nicole has quite the throwing arm! To figure out how fast she threw the ball, we need to use a little physics. When the ball reaches Chad's window, it's at the highest point in its trajectory, meaning its velocity is zero. So, let's focus on the ball coming back down.
Since Chad is 5.0m above the release point, we can say the height the ball fell is 5.0m. Now, we'll use a fun equation called the kinematic equation: v^2 = u^2 + 2gH.
In this equation, "v" represents the final velocity of the ball (which is 10m/s), "u" represents the initial velocity (the speed Nicole threw the ball), "g" is the acceleration due to gravity (9.8m/s^2 on Earth), and "H" is the height the ball fell (5.0m).
Plugging in the given values, we can solve for "u." Let's get those calculators ready! *beep boop beep boop*
10^2 = u^2 + 2(9.8)(5.0)
Simplifying that a bit, we have:
100 = u^2 + 98
Subtracting 98 from both sides, we get:
u^2 = 2
Well, that's convenient! To find the square root of 2, we ask a mathematician, but as a clown, I love twisting things around. So instead, let's round it up a bit and say Nicole threw the ball at approximately... √2 m/s. Now that's quite an interesting speed!
To find the speed at which Nicole threw the ball, we can use the concept of conservation of energy.
When the ball is at its maximum height, all of its initial kinetic energy is converted into potential energy.
Let's assume the initial speed with which Nicole throws the ball is v.
Using the conservation of energy, we can write:
Initial kinetic energy = Final potential energy
(1/2)mv^2 = mgh
where m is the mass of the ball, g is the acceleration due to gravity, and h is the maximum height the ball reaches.
Since we want to find the initial speed, we can rearrange the equation to solve for v:
v^2 = 2gh
Now, let's plug in the given values:
g = 9.8 m/s^2 (acceleration due to gravity)
h = 5.0 m (height difference between Nicole and Chad)
v^2 = 2 * 9.8 m/s^2 * 5.0 m
v^2 = 98 m^2/s^2
Taking the square root of both sides, we get:
v = √(98) m/s
v ≈ 9.899 m/s
Therefore, Nicole threw the ball with a speed of approximately 9.899 m/s.
To find the speed with which Nicole threw the ball, we can use the concept of conservation of energy. The total mechanical energy of the ball at the point of release is equal to the sum of its kinetic energy and potential energy.
When the ball is at its highest point, it has zero kinetic energy and maximum potential energy. At this point, all of the initial kinetic energy is converted into potential energy.
When the ball passes by Chad on the way up, we can find its potential energy and kinetic energy at that point. The total mechanical energy for the ball is the sum of the potential and kinetic energies.
In this case, the potential energy is mgh, where m is the mass of the ball, g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the ball above Chad (5.0m).
The kinetic energy can be calculated using the formula (1/2)mv^2, where m is the mass of the ball and v is its velocity (10 m/s in this case).
Since we know the total mechanical energy is conserved, we can set the initial mechanical energy equal to the mechanical energy at the point where Chad is observing the ball on its way up.
So, we have:
(mgh) + (1/2)mv^2 = (1/2)mv^2
Simplifying the equation, we get:
gh = 0
From here, we can see that h is equal to zero. This means that Chad is observing the ball at the same height at which Nicole threw it from.
Therefore, Nicole threw the ball from ground level (h = 0), and the initial potential energy of the ball is zero. Thus, the initial kinetic energy is equal to the total mechanical energy.
Using the formula (1/2)mv^2, we can now solve for the initial velocity (v) of the ball:
(1/2)mv^2 = (1/2)mv^2
The mass of the ball cancels out, leaving us with:
v = v
Therefore, the speed at which Nicole threw the ball is equal to the speed of the ball on its way down, 10 m/s.
V^2 = Vo^2 + 2g*d
Vo^2 = V^2 - 2g*d
Vo^2 = 10^2 - (-19.6)98*5
Vo^2 = 100 + 98 = 198
Vo = 14.1 m/s. = Speed at which ball was thrown.