Two blocks of masses m1=1.7 kg and m2=3.1 kg are put on a frictionless horizontal surface. A spring of stiffness k=950 N/m and negligible mass is placed between the blocks and they are pushed together to compress the spring by distance d=28 cm from its equilibrium length. The configuration is fixed by a cord that holds the blocks. When the cord is burned, the blocks move in opposite directions. Calculate the speed of the first block.
To calculate the speed of the first block, we need to consider the conservation of mechanical energy.
First, let's find the potential energy stored in the compressed spring:
Potential energy stored in a spring, PE = (1/2)kx^2
Here, k is the stiffness of the spring and x is the distance the spring is compressed from its equilibrium length. Given that k = 950 N/m and x = 28 cm = 0.28 m, we can calculate the potential energy:
PE = (1/2) * 950 N/m * (0.28 m)^2
PE = 18.424 N*m ≈ 18.42 J (to two decimal places)
According to the conservation of mechanical energy, this potential energy will be converted into the kinetic energy of the first block.
The kinetic energy of an object is given by the formula: KE = (1/2)mv^2
Where m is the mass of the object and v is its velocity.
Since there is no friction, all the potential energy from the spring will be converted into the kinetic energy of the first block.
Equating the potential energy to the kinetic energy, we have:
PE = KE
18.42 J = (1/2)m1v^2
Rearranging the equation, we can solve for v:
v = √((2*PE)/m1)
Substituting the given values, we get:
v = √((2*18.42 J)/(1.7 kg))
v ≈ √(21.65 m^2/s^2)
v ≈ 4.65 m/s (to two decimal places)
Therefore, the speed of the first block is approximately 4.65 m/s.