Compressed air is used to fire a 41.0 g ball vertically upward from a 1.0-m-tall tube. The air exerts an upward force of 3.7 N on the ball as long as it is in the tube. How high (in m) does the ball go above the top of the tube, assuming that the acceleration due to gravity is 9.80 m/s2?
To find the height the ball goes above the top of the tube, we can use the principle of conservation of energy. The initial potential energy of the ball in the tube is converted to kinetic energy as it moves upward against gravity. At the highest point, all the initial potential energy is converted back to potential energy.
Let's break down the steps to solve the problem:
Step 1: Calculate the initial potential energy in the tube.
The potential energy (PE) of an object near the Earth's surface is given by the equation PE = m * g * h, where m is the mass, g is the acceleration due to gravity, and h is the height.
In this case, the height is 1.0 m, so the initial potential energy is PE_initial = 41.0 g * 9.80 m/s^2 * 1.0 m.
Step 2: Calculate the work done by the compressed air.
The work (W) done by a force is given by the equation W = F * d, where F is the force and d is the displacement.
In this case, the force is 3.7 N, and the displacement is the height above the tube. Let's call it H. So, W = 3.7 N * H.
Step 3: Set up the conservation of energy equation.
Since the initial potential energy is converted into work done by the compressed air, we have PE_initial = W. From step 1 and 2, we can substitute the values and equations to get: 41.0 g * 9.80 m/s^2 * 1.0 m = 3.7 N * H.
Step 4: Solve for H.
Rearrange the equation from step 3 to solve for H: H = (41.0 g * 9.80 m/s^2 * 1.0 m) / (3.7 N).
Step 5: Calculate H.
Now, plug in the values and calculate H: H = (41.0 g * 9.80 m/s^2 * 1.0 m) / (3.7 N).
Using a calculator, the value of H is approximately 10.75 m.
Therefore, the ball goes approximately 10.75 meters above the top of the tube.
To find the height the ball reaches above the top of the tube, we can use the principles of projectile motion. We need to calculate the initial velocity of the ball first.
Step 1: Calculate the gravitational force on the ball:
The weight of the ball (force due to gravity) can be found using the formula:
Weight = mass * acceleration due to gravity
Weight = 41.0 g * 9.80 m/s^2 (convert grams to kilograms by dividing by 1000)
Weight = 0.0410 kg * 9.80 m/s^2
Weight = 0.4018 N
Step 2: Calculate the net force acting on the ball:
The upward force exerted by the compressed air is equal to the gravitational force acting down, so the net force is zero. This means the force of the air cancels out the force of gravity.
Step 3: Calculate the initial velocity of the ball:
The net force on the ball is equal to the mass of the ball multiplied by its acceleration.
Net Force = Mass * Acceleration
3.7 N = 0.0410 kg * Acceleration
Acceleration = 3.7 N / 0.0410 kg
Acceleration = 90.24 m/s^2
Step 4: Calculate the initial velocity (v₀) of the ball:
The final velocity (v) of the ball at maximum height is zero, so we can use the equation that relates initial velocity, final velocity, acceleration, and displacement:
v² = v₀² + 2ad
0 = v₀² + 2(-9.80 m/s²)(1 m)
0 = v₀² - 19.6 m²/s²
v₀² = 19.6 m²/s²
v₀ = √(19.6 m²/s²)
v₀ ≈ 4.43 m/s
Step 5: Calculate the maximum height (h) using the initial velocity and acceleration due to gravity:
Using the kinematic equation that relates displacement, initial velocity, acceleration, and time:
v = v₀ + at
0 = 4.43 m/s + (-9.80 m/s²) t (at maximum height, the final velocity is zero)
-4.43 m/s = -9.80 m/s² t
t ≈ 0.45 s
h = v₀t + (1/2)at² (equation of motion for displacement)
h = 4.43 m/s * 0.45 s + (1/2)(-9.80 m/s²)(0.45 s)²
h ≈ 1.98 m
Therefore, the ball reaches a height of approximately 1.98 meters above the top of the tube.
Set the work done on the ball by the gas equal maximum potential energy. Solve for the max. height, H.
3.7 N * 1.0 m = M*g*H
H = 3.7 J/(9.8*.041)= 9.2 m