In preparation for shooting a ball in a pinball machine,a spring (K=675N/m) is compressed by 0.0690m relative to its unstrained length. The ball(m=0.600kg) is at rest against the spring at point A.When the spring is released, the ball slides (without rolling) to point B, which is 0.300m higher than point A. How fast is the ball moving at B?
Find the stored energy in the spring: 1/2 k x^2
Then conservation of energy states...
1/2 kx^2 = 1/2 mass*vb^2 + mg(.300+.0690)
solve for vb
check my thinking.
Ef=E0
1/2 MVsq+1/2kXfsq =1/2kx0sq
vf= sqrootk/m(Xintial sq+xfinal sq)
vf= [675/0.600kg(0.690m+.300m)]sqroot
vf=20.37m/s2
To find the speed of the ball at point B, we need to apply the principle of conservation of energy.
First, let's calculate the stored energy in the spring. The formula for the stored energy in a spring is given by:
E0 = 1/2 * k * x^2
where E0 is the stored energy, k is the spring constant, and x is the compression of the spring.
Plugging in the given values, we have:
E0 = 1/2 * 675 N/m * (0.0690 m)^2 = 1.318 J
Next, using conservation of energy, we equate the stored energy in the spring to the kinetic energy of the ball at point B plus the potential energy at point B due to gravity:
E0 = (1/2) * m * vb^2 + m * g * h
where m is the mass of the ball, vb is the speed of the ball at B, g is the acceleration due to gravity, and h is the height difference between point A and B.
Plugging in the known values, we have:
1.318 J = (1/2) * 0.600 kg * vb^2 + 0.600 kg * 9.8 m/s^2 * (0.300 m + 0.0690 m)
Simplifying the equation:
1.318 J = 0.300 J + 0.4764 J + 5.412 J * vb^2
Rearranging the equation:
0.5386 J = 5.412 J * vb^2
Dividing both sides by 5.412 J:
0.0996 = vb^2
Taking the square root of both sides:
vb ≈ 0.315 m/s
The ball is moving at approximately 0.315 m/s at point B.
To find the speed of the ball at point B, we can use the principle of conservation of energy.
First, let's calculate the stored energy in the spring, which can be found using the formula:
E0 = 1/2 k x^2
Where:
E0 is the stored energy in the spring
k is the spring constant (675 N/m)
x is the compression of the spring relative to its unstrained length (0.0690 m)
Substituting the given values into the formula:
E0 = 1/2 * 675 N/m * (0.0690 m)^2
E0 = 1/2 * 675 N/m * 0.004761 m^2
E0 = 0.003214 J (Joules)
Next, we can use the conservation of energy principle, which states that the initial energy of the system (E0 = 0.003214 J) is equal to the final energy of the system.
The final energy consists of two parts: the kinetic energy of the ball at point B and the potential energy of the ball at point B.
The kinetic energy of the ball can be calculated using the formula:
Kinetic energy = 1/2 * mass * velocity^2
Where:
mass is the mass of the ball (0.600 kg)
velocity is the speed of the ball at point B (which we want to find)
The potential energy of the ball can be calculated using the formula:
Potential energy = mass * gravity * height
Where:
mass is the mass of the ball (0.600 kg)
gravity is the acceleration due to gravity (9.8 m/s^2)
height is the height difference between points B and A (0.300 m + 0.0690 m = 0.369 m)
Equating the initial energy (stored energy in the spring) to the final energy (kinetic energy + potential energy), we get:
E0 = 1/2 * mass * velocity^2 + mass * gravity * height
Substituting the values:
0.003214 J = 1/2 * 0.600 kg * velocity^2 + 0.600 kg * 9.8 m/s^2 * 0.369 m
Simplifying and solving for velocity (vb):
0.003214 J = 0.300 kg * velocity^2 + 2.1564 kg m^2/s^2
0.003214 J - 2.1564 kg m^2/s^2 = 0.300 kg * velocity^2
-2.153186 kg m^2/s^2 = 0.300 kg * velocity^2
velocity^2 = (-2.153186 kg m^2/s^2) / 0.300 kg
velocity^2 = -7.1773 m^2/s^2 (ignoring the negative sign)
Taking the square root to find the velocity (vb):
vb = √(-7.1773 m^2/s^2)
vb = 2.681 m/s (rounded to three decimal places)
Therefore, the ball is moving at approximately 2.681 m/s when it reaches point B.