A ball of mass 522 g starts at rest and slides down a frictionless track, as shown below. It leaves the track horizontally, striking the ground a distance x = 0.93 m from the end of the track after falling a vertical distance h2 = 1.18 m from the end of the track.
(a) At what height above the ground does the ball start to move?
(b) What is the speed of the ball when it leaves the track?
(c) What is the speed of the ball when it hits the ground?
To answer these questions, we can use the principles of conservation of energy. The total mechanical energy of the system (ball and Earth) is conserved, so we can equate the initial and final energies.
(a) To find the height above the ground where the ball starts to move, we need to find the potential energy at that point. The initial mechanical energy of the system is given by the potential energy at the starting height, which can be calculated using the equation:
Potential energy (P.E.) = mass * gravitational acceleration * height
P.E. = (0.522 kg) * (9.8 m/s^2) * h1
The final mechanical energy is the sum of the kinetic energy and potential energy at the point where the ball leaves the track. The kinetic energy can be calculated using the equation:
Kinetic energy (K.E.) = 0.5 * mass * velocity^2
Let's denote the final velocity by v and the final height above the ground by h2. The final mechanical energy can be written as:
Final mechanical energy = 0.5 * (0.522 kg) * v^2 + (0.522 kg) * (9.8 m/s^2) * h2
Since the total mechanical energy is conserved, we can equate the initial and final mechanical energies:
(0.522 kg) * (9.8 m/s^2) * h1 = 0.5 * (0.522 kg) * v^2 + (0.522 kg) * (9.8 m/s^2) * h2
Simplifying the equation and solving for h1, we get:
h1 = 0.5 * v^2 + h2
Now, we need to find the value of v and h2 to calculate h1 and find the height above the ground where the ball starts to move.
(b) To find the speed of the ball when it leaves the track, we need to use the conservation of mechanical energy. The initial mechanical energy (before leaving the track) is equal to the final mechanical energy (at the point of leaving).
The initial mechanical energy consists of only potential energy. The initial potential energy is given by the equation:
Potential energy = mass * gravitational acceleration * height
The final mechanical energy consists of kinetic energy (0.5 * mass * velocity^2) and potential energy (mass * gravitational acceleration * h2).
Equating the initial and final mechanical energies, we can solve for the final velocity (v).
(c) To find the speed of the ball when it hits the ground, we can use the principle of conservation of energy. The initial mechanical energy (before leaving the track) is equal to the final mechanical energy (when the ball hits the ground).
The initial mechanical energy consists of potential energy (mass * gravitational acceleration * h1) only.
The final mechanical energy consists of kinetic energy (0.5 * mass * velocity^2) and potential energy (mass * gravitational acceleration * 0).
Equating the initial and final mechanical energies, we can solve for the final velocity (v).
To answer these questions, we can use the principles of conservation of energy and projectile motion.
(a) To calculate the height above the ground where the ball starts to move, we need to find the potential energy at that point.
The potential energy of an object is given by the equation: PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
Since the ball is at rest, it has no kinetic energy, so all its energy is potential energy. Therefore, at the starting point, the potential energy is equal to the total energy, which is given by mgh2.
We can substitute the known values into the equation:
PE = (0.522 kg) * (9.8 m/s^2) * (1.18 m) = 6.0948 J
So, the ball starts to move when the height above the ground has a potential energy of 6.0948 J.
(b) To find the speed of the ball when it leaves the track, we can use the principle of conservation of mechanical energy.
The mechanical energy of an object is the sum of its kinetic and potential energy.
At the starting point, the ball has no kinetic energy (as it is at rest), so all its energy is potential energy.
At the end point, the ball has no potential energy (as it is at ground level), so all its energy is kinetic energy.
According to the conservation of mechanical energy, the initial potential energy is equal to the final kinetic energy.
PE = KE
mgh2 = (1/2)mv^2
Where v is the velocity of the ball when it leaves the track.
By rearranging the equation and solving for v, we can find the speed:
v = sqrt(2gh2)
Substituting the known values:
v = sqrt(2 * 9.8 m/s^2 * 1.18 m) = 5.172 m/s
So the speed of the ball when it leaves the track is 5.172 m/s.
(c) To find the speed of the ball when it hits the ground, we can use projectile motion equations.
The horizontal distance traveled by the ball (x) is given, and we are trying to find its initial vertical velocity (Vy).
The horizontal distance is given by the equation: x = Vx * t, where Vx is the horizontal velocity of the ball and t is the time of flight.
Since the track is frictionless, there are no horizontal forces acting on the ball, so the horizontal velocity remains constant.
Therefore, we can write: Vx = v (from part b)
The time of flight can be found using the equation: h2 = (1/2)gt^2, where h2 is the vertical distance and g is the acceleration due to gravity.
Solving for t, we get: t = sqrt((2 * h2) / g)
Substituting the known values:
t = sqrt((2 * 1.18 m) / 9.8 m/s^2) = 0.504 s
Now, we can find the initial vertical velocity (Vy) using the equation: Vy = gt.
Vy = 9.8 m/s^2 * 0.504 s = 4.9392 m/s
Finally, we can find the total speed (V) when the ball hits the ground using the Pythagorean theorem: V = sqrt(Vx^2 + Vy^2)
V = sqrt((5.172 m/s)^2 + (4.9392 m/s)^2) = 7.015 m/s
So the speed of the ball when it hits the ground is 7.015 m/s.