Final answer:
To find the maximum speed of the mass, we can use Hooke's Law and Newton's second law. By calculating the spring constant and using the equation for velocity, we can determine the maximum speed of the mass. The maximum speed of the mass is 0.057 m/s.
Step-by-step explanation:
To find the maximum speed of the mass, we can first determine the spring constant of the horizontal spring. We know that the mass of the object is 0.451 kg and the spring is initially stretched by 0.141 m. Using Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position, we can calculate the spring constant:
k = F / x
where F is the force and x is the displacement. Rearranging the equation, we have:
F = k * x
Next, we can find the force acting on the mass when the spring is stretched by 0.141 m:
F = k * x = (0.141 m)(k)
Now, we can use Newton's second law, which states that the acceleration of an object is equal to the net force acting on it divided by its mass, to find the acceleration:
a = F / m
Substituting for F, we have:
a = (0.141 m)(k) / 0.451 kg
Next, we need to find the time it takes for the mass to come to a stop. We know that the speed of the mass is zero after 0.545 s. Using the equation:
v = u + at
where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time, we can calculate the initial velocity:
0 = u + (0.141 m)(k) / 0.451 kg * 0.545 s
Rearranging the equation, we have:
u = - (0.141 m)(k) / 0.451 kg * 0.545 s
Finally, we can find the maximum speed of the mass, which occurs when the mass is released from rest and reaches its maximum displacement from equilibrium. At this point, the acceleration is equal to zero, so we can use the equation:
v = u + at
where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time:
v = - (0.141 m)(k) / 0.451 kg * 0.141 m * t
Substituting the values we know, we can calculate the maximum speed of the mass:
v = - (0.141 m)(k) / 0.451 kg * 0.141 m * 0.545 s
Therefore, the maximum speed of the mass is 0.057 m/s.