A 0.103 kg ball is moving 11.9 m/s when it runs into a spring of spring constant 66.9 N/M . How far does the spring compress in bringing the ball to a stop?
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Sure, I'd be happy to help! To determine how far the spring compresses in bringing the ball to a stop, we can use the principle of conservation of mechanical energy.
The total mechanical energy of the ball-spring system is conserved, meaning the initial kinetic energy of the ball is converted to potential energy stored in the spring, as the ball comes to a stop.
The formula for the potential energy stored in a spring is given by:
PE = (1/2)kx^2
where PE is the potential energy, k is the spring constant, and x is the displacement or compression of the spring.
Initially, the ball has kinetic energy given by:
KE = (1/2)mv^2
where KE is the kinetic energy, m is the mass of the ball, and v is the velocity of the ball.
Since the total mechanical energy is conserved, we can equate the initial kinetic energy to the final potential energy:
(1/2)mv^2 = (1/2)kx^2
Now we can solve for x, the compression of the spring:
x^2 = (mv^2) / k
Substituting the given values:
m = 0.103 kg
v = 11.9 m/s
k = 66.9 N/m
x^2 = (0.103 kg × (11.9 m/s)^2) / 66.9 N/m
Calculating the value in parentheses:
x^2 = 0.103 kg × 141.61 m^2/s^2 / 66.9 N/m
x^2 = 0.21749837 m^2
Taking the square root of both sides to solve for x:
x ≈ sqrt(0.21749837 m^2)
x ≈ 0.466 m
Therefore, the spring compresses approximately 0.466 meters in bringing the ball to a stop.
To find the distance the spring compresses, we need to use the principle of conservation of energy. The initial kinetic energy of the ball will be converted into potential energy stored in the spring when it comes to a stop.
The equation for the potential energy stored in a spring is given by:
Potential Energy (PE) = (1/2) * k * x^2,
where k is the spring constant and x is the displacement (compression) of the spring.
The initial kinetic energy of the ball can be calculated using the equation:
Initial Kinetic Energy (KE) = (1/2) * mass * velocity^2.
Since conservation of energy tells us that the initial kinetic energy is equal to the potential energy when the ball stops, we can equate the two expressions:
(1/2) * mass * velocity^2 = (1/2) * k * x^2.
Plugging in the given values:
mass = 0.103 kg,
velocity = 11.9 m/s, and
k = 66.9 N/M,
we can solve for x:
(1/2) * 0.103 kg * (11.9 m/s)^2 = (1/2) * 66.9 N/M * x^2.
Simplifying the equation:
0.065095 kg * m^2/s^2 = 33.45 N/M * x^2.
Dividing both sides by 33.45 N/M, we get:
(0.065095 kg * m^2/s^2) / (33.45 N/M) = x^2.
Taking the square root of both sides, we find that:
x = √[(0.065095 kg * m^2/s^2) / (33.45 N/M)].
Evaluating the expression:
x = √[0.0019446 m^2] (rounded to 7 decimal places),
x ≈ 0.0441 m.
Therefore, the spring compresses by approximately 0.0441 meters in bringing the ball to a stop.
the kinetic energy of the ball is the energy that compresses the spring
1/2 m v^2 = 1/2 k x^2
.103 * 11.9^2 = 66.9 * x^2