The ball launcher in a pinball machine has a spring that has a force constant of 1.50 N/cm.
The surface on which the ball moves is inclined a = 14.5o with respect to the horizontal. If the spring is initially compressed 4.70 cm, find the launching speed of a 0.113 kg ball when the plunger is released. Friction and the mass of the plunger are negligible.
If we assume the mass of the ball is proportional to how many significant digits there are in the problem (3), we can assume the mass is around 3, using the formula Ek = m/2v^2 we can find the answer to be -8.
To find the launching speed of the ball, we can use the principle of conservation of mechanical energy. The initial potential energy stored in the compressed spring is converted into kinetic energy when the ball is released.
1. First, let's calculate the potential energy stored in the compressed spring:
Potential energy (PE) = (1/2) * k * x^2
where k is the spring constant and x is the compression distance.
Given:
k = 1.50 N/cm = 1.50 N / 0.01 m (since 1 cm = 0.01 m)
x = 4.70 cm = 4.70 * 0.01 m
PE = (1/2) * (1.50 N / 0.01 m) * (4.70 * 0.01 m)^2
= (1/2) * (150 N/m) * (0.047 m)^2
= 0.1659 J (rounded to four decimal places)
2. Now, we know that the potential energy is converted into kinetic energy when the ball is released:
PE = KE
KE = (1/2) * m * v^2
where m is the mass of the ball and v is the launching speed.
Given:
m = 0.113 kg
0.1659 J = (1/2) * (0.113 kg) * v^2
v^2 = (2 * 0.1659 J) / (0.113 kg)
v^2 = 2.9314 J/kg
v = √(2.9314 J/kg)
v = 1.71 m/s (rounded to two decimal places)
Therefore, the launching speed of the 0.113 kg ball is approximately 1.71 m/s.
To find the launching speed of the ball, we can use the concept of conservation of mechanical energy. The mechanical energy of the ball is conserved when the spring is released and the ball is launched.
The mechanical energy before the launch is given by the potential energy stored in the compressed spring. The potential energy stored in a spring is given by the equation:
PE = (1/2) * k * x^2,
where PE is the potential energy, k is the force constant of the spring, and x is the compression or extension of the spring.
In this case, the potential energy is:
PE = (1/2) * (1.50 N/cm) * (4.70 cm)^2.
Next, we need to calculate the change in height of the ball as it moves up the inclined plane. The change in height is given by:
Δh = x * sin(a),
where a is the angle of inclination and x is the compression or extension of the spring.
In this case, the change in height is:
Δh = (4.70 cm) * sin(14.5o).
To find the launching speed, we can equate the potential energy to the change in kinetic energy. The kinetic energy of the ball is given by:
KE = (1/2) * m * v^2,
where KE is the kinetic energy, m is the mass of the ball, and v is the velocity of the ball.
Setting the potential energy equal to the change in kinetic energy, we have:
(1/2) * (1.50 N/cm) * (4.70 cm)^2 = (1/2) * (0.113 kg) * v^2.
Simplifying the equation, we have:
(1.50 N/cm) * (4.70 cm)^2 = (0.113 kg) * v^2.
Solving for v, we find:
v^2 = [(1.50 N/cm) * (4.70 cm)^2] / (0.113 kg).
Taking the square root of both sides, we can find the launching speed of the ball:
v = √{[(1.50 N/cm) * (4.70 cm)^2] / (0.113 kg)}.