Alexis D. answered • 01/21/23
Neuroscience PhD Specializing in Math, Bio, and Test Prep
When the block is released, it will rise vertically because it will have gained potential energy from compressing the spring. The potential energy stored in a spring is given by the equation:
U = 1/2 kx^2
where U is the potential energy, k is the spring constant, and x is the amount of compression or stretching.
In this case, the spring constant is 5.00 ✕ 103 N/m and the compression of the spring is 0.090 m. So the potential energy stored in the spring is:
U = 1/2 (5.00 ✕ 103 N/m)(0.090 m)^2 = 2.025 J
When the block is released, it will convert the potential energy stored in the spring into kinetic energy. The total energy of the system is conserved, so the kinetic energy of the block when it leaves the spring is also 2.025 J.
The kinetic energy of an object is given by the equation:
K = 1/2 mv^2
where K is the kinetic energy, m is the mass of the object, and v is the velocity of the object.
Solving for the velocity of the block, we get:
v = sqrt(2K/m) = sqrt(2*2.025 J/0.280 kg) = sqrt(14.32142857) = 3.80 m/s
To find how high the block rises above the point of release, we use the relationship between the initial velocity, the final velocity and the time taken to reach the peak:
h = v^2 / 2g
where h is the height, v is the final velocity, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
The final velocity is 3.80 m/s, so the height above the point of release is:
h = (3.80^2) / (2*9.8) = 14.44/19.6 = 0.74 m
The block will rise 0.74m above the point of release.
