A stone with mass m = 2.3 kg is thrown vertically upward into the air with an initial kinetic energy of 190 J. The drag force acting on the stone throughout its flight is constant, independent of the velocity of the stone, and has a magnitude of 2.3 N. What is the maximum height reached by the stone?
What is the speed of the stone upon impact with the ground?
2.3
A stone thrown up passes same height "h" at times 2.5s and 8.0s (take g=9.8)
To find the maximum height reached by the stone, we need to use the principle of conservation of energy.
Step 1: Determine the initial potential energy of the stone
Given that the initial kinetic energy is 190 J, we can assume that all of this kinetic energy will be converted into potential energy at the maximum height. So, the initial potential energy is also 190 J.
Step 2: Determine the work done by the drag force
The work done by the drag force can be calculated using the formula:
Work = Force x Distance x cos(theta)
where theta is the angle between the force vector and the displacement vector. In this case, since the stone is traveling vertically upwards, the angle between the force and displacement is 0 degrees. So, cos(0) = 1.
The work done by the drag force is equal to the force multiplied by the displacement. The force is 2.3 N and the displacement is the same as the distance traveled by the stone. We need to calculate the distance traveled by the stone to find the work done by the drag force.
Step 3: Calculate the distance traveled by the stone
To calculate the distance traveled, we need to consider the initial velocity and acceleration of the stone. The initial velocity is given as the velocity when launched. The acceleration can be calculated using Newton's second law: F = ma. In this case, the net force acting on the stone is the difference between the force of gravity and the drag force since the stone is moving vertically upwards.
Net force = mg - drag force
Net force = 2.3 kg x 9.8 m/s^2 - 2.3 N
Step 4: Calculate the acceleration of the stone
Using Newton's second law, we can find the acceleration of the stone:
F = ma
2.3 kg x 9.8 m/s^2 - 2.3 N = 2.3 kg x a
Step 5: Solve for the acceleration
Solving for a:
a = (2.3 kg x 9.8 m/s^2 - 2.3 N) / 2.3 kg
Step 6: Calculate the time of flight
The time taken for the stone to reach maximum height and then fall back down can be calculated using the equation of motion for vertical motion:
s = ut + 0.5at^2
where s is the distance traveled, u is the initial velocity, a is the acceleration, and t is the time taken.
In this case, the distance traveled is 0 since the stone returns to its starting point. The initial velocity is the same as the final velocity when it reaches maximum height and starts falling. So, we can calculate the time of flight using the equation:
0 = 0.5at^2
t = sqrt(2s / a)
Step 7: Calculate the distance traveled
The distance traveled can be calculated using the equation:
s = ut + 0.5at^2
Since the stone is moving vertically upwards, the initial velocity is positive. The displacement or distance traveled is the same as the distance to maximum height, which is what we need to find.
So, the equation becomes:
0 = (u)(t) + 0.5(-a)(t^2)
t = 2u / a
Substituting the expression for acceleration from Step 5 into the equation:
t = 2u / [(2.3 kg x 9.8 m/s^2 - 2.3 N) / 2.3 kg]
Step 8: Calculate the distance traveled (continued)
Substituting the expression for time from Step 7 into the equation:
s = (u)(t) + 0.5(-a)(t^2)
s = (u)(2u / a) + 0.5(-a)((2u / a)^2)
s = 2u^2/a - (2u^2/a)
s = 0
So, the distance traveled by the stone is 0.
Step 9: Calculate the work done by the drag force
Since the distance traveled is 0, the work done by the drag force is also 0, as work is equal to force multiplied by displacement. Therefore, no work is done by the drag force on the stone.
Step 10: Calculate the final potential energy of the stone
At maximum height, all the initial kinetic energy is converted into potential energy. So, the final potential energy is equal to the initial kinetic energy, which is 190 J.
Step 11: Calculate the maximum height
The maximum height can be calculated using the equation:
Potential energy = mgh
where m is the mass of the stone, g is the acceleration due to gravity, and h is the maximum height.
Substituting the values:
190 J = 2.3 kg x 9.8 m/s^2 x h
Solving for h:
h = 190 J / (2.3 kg x 9.8 m/s^2)
Step 12: Calculate the speed of the stone upon impact with the ground
To calculate the speed of the stone upon impact with the ground, we can use the principle of conservation of energy. Since no work is done by the drag force, the total mechanical energy of the stone is conserved. The total mechanical energy is the sum of the initial kinetic energy and the initial potential energy, which is also equal to the final kinetic energy at impact.
Total mechanical energy = Initial kinetic energy + Initial potential energy = Final kinetic energy
The final kinetic energy can be calculated using the equation:
Final kinetic energy = 0.5mv^2
where m is the mass of the stone and v is the speed of the stone upon impact.
Substituting the values:
190 J + 0 J = 0.5 x 2.3 kg x v^2
Solving for v:
v^2 = (190 J / (0.5 x 2.3 kg))
v = sqrt(190 J / (0.5 x 2.3 kg))
So, the maximum height reached by the stone is (190 J / (2.3 kg x 9.8 m/s^2)) and the speed of the stone upon impact with the ground is sqrt(190 J / (0.5 x 2.3 kg)).
To find the maximum height reached by the stone, we can use the principle of conservation of energy. The initial kinetic energy of the stone is given as 190 J. This energy will be converted into potential energy at the maximum height. We can calculate the maximum height using the following equation:
Potential Energy = m * g * h,
where m is the mass of the stone, g is the acceleration due to gravity, and h is the maximum height.
The potential energy at the maximum height is equal to the initial kinetic energy:
m * g * h = 190 J.
Rearranging the equation, we get:
h = 190 J / (m * g).
We know that the mass of the stone is 2.3 kg and the acceleration due to gravity is approximately 9.8 m/s^2. Plugging these values into the equation, we can solve for h:
h = 190 J / (2.3 kg * 9.8 m/s^2) ≈ 8.07 m.
Therefore, the maximum height reached by the stone is approximately 8.07 meters.
To find the speed of the stone upon impact with the ground, we need to consider the loss of potential energy and the work done by the drag force. At the maximum height, all the potential energy is converted into kinetic energy. The work done by the drag force is equal to the force multiplied by the displacement.
The work done by the drag force is given by:
Work = Drag force * displacement.
Since the drag force is constant and independent of the velocity, the work done is:
Work = Drag force * (2 * maximum height),
where the factor of 2 accounts for the stone's upward and downward motion.
The loss of potential energy is equal to the initial potential energy minus the potential energy at the maximum height:
Loss of Potential Energy = m * g * h.
The loss of potential energy will be equal to the work done by the drag force, so we can equate the two:
Drag force * (2 * maximum height) = m * g * h.
Rearranging the equation, we can solve for the maximum height:
maximum height = (m * g * h) / (2 * Drag force).
Plugging in the given values, we get:
maximum height = (2.3 kg * 9.8 m/s^2 * 8.07 m) / (2 * 2.3 N) ≈ 11.05 m.
Therefore, the maximum height reached by the stone is approximately 11.05 meters.
To find the speed of the stone upon impact with the ground, we can use the principle of conservation of mechanical energy. At the maximum height, we can equate the initial kinetic energy (190 J) to the sum of the final kinetic energy and the work done by the drag force during descent:
190 J = (1/2) * m * v^2 + (Drag force * displacement).
Since the stone impacts the ground, its displacement is equal to the maximum height (11.05 m). The drag force is constant throughout the stone's flight and is given as 2.3 N.
Rearranging the equation, we can solve for the final velocity (v) upon impact:
v^2 = (2 * (190 J - Drag force * displacement)) / m.
Plugging in the given values, we get:
v^2 = (2 * (190 J - 2.3 N * 11.05 m)) / 2.3 kg ≈ 35.99 m^2/s^2.
Taking the square root of both sides, we find:
v ≈ √(35.99 m^2/s^2) ≈ 6.00 m/s.
Therefore, the speed of the stone upon impact with the ground is approximately 6.00 meters per second.