A ball of mass m = 0.75 kg is thrown straight upward with an initial speed of 8.9 m/s. Plot the gravitational potential energy of the block from its launch height, y = 0, to the height y = 5.0 m. Let U = 0 correspond to y = 0. Determine the turning point (maximum height) of this mass.
Vf^2 = Vo^2 + 2gd = 0.
(8.9)^2 + 2(-9.8)d = 0,
79.21 - 19.6d = 0,
-19.6d = -79.21,
d(max) = -79.21 / -19.6 = 4.04m.
Well, well, well! It seems we have a ball on the move here, and we need to keep track of its potential energy. How exciting! Let's get started, shall we?
Now, when the ball is at its launch height, y = 0, its gravitational potential energy, U, is also equal to 0. As the ball moves upward, its potential energy starts to increase.
The gravitational potential energy, U, can be calculated using the formula U = mgh, where m is the mass of the ball, g is the acceleration due to gravity, and h is the height at which we need to determine the potential energy.
First, let's find the potential energy when the ball is at height y = 5.0 m. We have m = 0.75 kg and g is approximately 9.8 m/s^2. So, plugging in the numbers:
U = (0.75 kg) * (9.8 m/s^2) * (5.0 m)
Calculating that gives us U = 36.75 Joules.
Now, to find the turning point, we want to determine the maximum height that the ball reaches. At the turning point, the ball's velocity will be zero momentarily. That's where the ball reaches its highest point and gravity starts pulling it back down.
Using the energy conservation principle, we can say that the initial kinetic energy of the ball will be converted to potential energy at the maximum height.
So, let's find the initial kinetic energy of the ball using the formula:
K.E. = (1/2) * m * v^2
Here, m is the mass of the ball and v is its initial velocity, which is given as 8.9 m/s. Plugging in the values:
K.E. = (1/2) * (0.75 kg) * (8.9 m/s)^2
Calculating that gives us K.E. = 29.87 Joules.
Since energy is conserved, the initial kinetic energy (K.E.) is equal to the final potential energy (U) at the maximum height.
Therefore, 29.87 Joules = U.
Now we can go back to the formula of gravitational potential energy, U = mgh. We can rearrange it to solve for h:
h = U / (mg)
Substituting the values, we have:
h = 29.87 Joules / ((0.75 kg) * (9.8 m/s^2))
Calculating that gives us h = 4.02 meters.
So, the turning point (maximum height) of the ball is approximately 4.02 meters.
Oh, what a journey for that little ball! Just remember, gravity may bring us down, but it also lifts us up, and all the potential energy we gain is truly uplifting... in a scientific way, of course!
To plot the gravitational potential energy of the ball as it moves from y = 0 to y = 5.0 m, we need to calculate the gravitational potential energy at different heights.
Gravitational potential energy can be calculated using the equation:
U = m * g * h
Where U is the gravitational potential energy, m is the mass of the ball, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height.
Let's calculate the gravitational potential energy at y = 0 and y = 5.0 m.
At y = 0:
U₀ = m * g * h₀
= 0.75 kg * 9.8 m/s² * 0
= 0 J
At y = 5.0 m:
U₁ = m * g * h₁
= 0.75 kg * 9.8 m/s² * 5.0 m
= 36.75 J
Now, let's plot the gravitational potential energy on a graph with height (y) on the x-axis and potential energy (U) on the y-axis. We will plot points at y = 0 and y = 5.0 m.
Height (y) | Potential Energy (U)
--------------------------------
0 m | 0 J
5.0 m | 36.75 J
To determine the turning point or maximum height of the ball, we need to find the point on the graph where the potential energy is highest.
In this case, the maximum height is y = 5.0 m, where the potential energy is 36.75 J. This is the turning point for this mass.
To plot the gravitational potential energy of the ball as a function of height, we need to use the formula:
U = m * g * h
where U is the gravitational potential energy, m is the mass of the ball, g is the acceleration due to gravity, and h is the height.
In this case, the ball is thrown straight upward, so the acceleration due to gravity is directed downwards and has a value of approximately 9.8 m/s^2.
To determine the turning point (maximum height) of the ball, we can use the kinematic equation for vertical motion:
v_f^2 = v_i^2 + 2 * a * d
where v_f is the final velocity, v_i is the initial velocity, a is the acceleration, and d is the displacement.
At the maximum height, the final velocity is zero, and the acceleration is -9.8 m/s^2 (opposite to the direction of the initial velocity). We can rearrange the equation to solve for the displacement:
d = (v_f^2 - v_i^2) / (2 * a)
Substituting in the given values:
d = (0^2 - (8.9 m/s)^2) / (2 * (-9.8 m/s^2))
Simplifying:
d = (-79.21 m^2/s^2) / (-19.6 m/s^2)
d = 4.04 m
Therefore, the turning point (maximum height) of the ball is 4.04 meters.