I really need help on D part of the following problem:
A clever engineer designs a "sprong" that obeys the force law F subscript e = -q(x - x subscript e)^3, where x subscript e is the equilibrium position of the end of the sprong and q is the sprong constant. For simplicity, x subscript e = 0m . Then F subscript x = -qx^3.
Part A:
What are the units of q? N/m^3 (correct)
Part B:
graph
Part C:
Find an expression for the potential energy of a stretched or compressed sprong. U(x) = 1/4qx^4 (correct)
Part D:
A sprong-loaded toy gun shoots a 24 g plastic ball. What is the launch speed if the sprong constant is 4.3 × 10^4 N/m^3, with the units you found in part A, and the sprong is compressed 11 cm? Assume the barrel is frictionless.
KE=Integral force.dx=1/4q x^4
1/2 m v^2=1/4 4.3E4N/m^4 (.11)^4
solve for v
Put mass in kilogram units.
Well, it looks like you've made it to part D, the last and final part of the problem. Don't worry, we're almost done!
Given the sprong constant (q) of 4.3 × 10^4 N/m^3 and the sprong being compressed 11 cm, we can use the expression for potential energy (U(x) = 1/4qx^4) we derived in part C.
Since we want to find the launch speed of the plastic ball, we can equate the potential energy to the kinetic energy of the ball when it is fully released.
The potential energy at the compressed position is U(x) = 1/4(4.3 × 10^4 N/m^3)(0.11 m)^4. And since the ball is at the compressed position, all of the potential energy will be converted into kinetic energy.
So, we can write: 1/4(4.3 × 10^4 N/m^3)(0.11 m)^4 = 1/2m(v)^2, where m is the mass of the ball and v is the launch speed we want to find.
You mentioned that the ball weighs 24 grams, which is equivalent to 0.024 kg. Plugging in the values, we have:
1/4(4.3 × 10^4 N/m^3)(0.11 m)^4 = 1/2(0.024 kg)(v)^2
Now, let me just grab my calculator and do some quick calculations...
*beep boop beep*
After crunching the numbers, I got v = 3.72 m/s. So, the launch speed of the plastic ball from the sprong-loaded toy gun is approximately 3.72 m/s.
And there you have it! Problem solved. I hope that puts a spring in your step!
To find the launch speed of the sprong-loaded toy gun, we can use the principle of conservation of mechanical energy. When the sprong is compressed, it stores potential energy, which gets converted into kinetic energy when the sprong is released and the ball is launched.
Step 1: Convert the mass from grams to kilograms.
Given that the mass of the plastic ball is 24 g, we need to convert it to kilograms by dividing by 1000:
Mass (m) = 24 g / 1000 = 0.024 kg
Step 2: Convert the sprong constant to SI units.
Given that the sprong constant is 4.3 × 10^4 N/m^3, we need to convert it to N/m:
Sprong constant (q) = 4.3 × 10^4 N/m^3
Step 3: Convert the compression distance to meters.
Given that the sprong is compressed by 11 cm, we need to convert it to meters by dividing by 100:
Compression distance (x) = 11 cm / 100 = 0.11 m
Step 4: Calculate the potential energy of the compressed sprong.
The potential energy of a compressed sprong can be found using the expression U(x) = 1/4qx^4, where x is the compression distance and q is the sprong constant:
U(x) = 1/4 * (4.3 × 10^4 N/m^3) * (0.11 m)^4
Step 5: Calculate the kinetic energy at the launch.
Since mechanical energy is conserved, the potential energy of the compressed sprong (U(x)) gets converted into kinetic energy when the sprong is released. At the launch, all potential energy is converted into kinetic energy, so we can equate them:
U(x) = 1/2 * mv^2
Step 6: Solve for the launch speed (v).
Substituting the values into the equation:
1/4 * (4.3 × 10^4 N/m^3) * (0.11 m)^4 = 1/2 * (0.024 kg) * v^2
Simplifying and solving for v:
(4.3 × 10^4 N/m^3) * (0.11 m)^4 = 2 * (0.024 kg) * v^2
v^2 = [(4.3 × 10^4 N/m^3) * (0.11 m)^4] / [2 * (0.024 kg)]
v^2 ≈ 5.404 N/m^3 * m^4 / kg
Taking the square root of both sides:
v ≈ √(5.404 N/m^3 * m^4 / kg)
v ≈ 2.33 m/s (rounded to two decimal places)
Therefore, the launch speed of the sprong-loaded toy gun is approximately 2.33 m/s.
To find the launch speed of the plastic ball in the given scenario, we can use the principles of potential and kinetic energy.
First, let's determine the potential energy stored in the compressed sprong. We know from Part C that the expression for potential energy is given by U(x) = 1/4qx^4, where q is the sprong constant.
Given that the sprong constant is 4.3 × 10^4 N/m^3, and the sprong is compressed 11 cm, we need to convert the distance to meters.
11 cm = 0.11 m
Now, let's substitute the values into the formula:
U(x) = 1/4 * (4.3 × 10^4 N/m^3) * (0.11 m)^4
Simplifying this expression will give us the potential energy of the compressed sprong.
After the sprong is released, the potential energy is converted into the kinetic energy of the plastic ball. The expression for kinetic energy is given by:
KE = 1/2 * m * v^2
where m is the mass of the plastic ball and v is the launch speed we want to find.
Given that the mass of the ball is 24 g, we need to convert grams to kilograms.
24 g = 0.024 kg
Now, we can equate the potential energy of the compressed sprong to the kinetic energy of the ball:
1/4 * (4.3 × 10^4 N/m^3) * (0.11 m)^4 = 1/2 * (0.024 kg) * v^2
Rearranging the equation for v, we get:
v = √(2 * 1/4 * (4.3 × 10^4 N/m^3) * (0.11 m)^4 / (0.024 kg))
Simplifying further will give us the launch speed of the plastic ball in meters per second (m/s).
Remember to substitute the values correctly, perform the calculations step by step, and pay attention to units in order to arrive at the correct answer.