Toolkit
Block A slides down a ramp whose slope is 45°. An identical block B slides down a ramp whose slope is 60°. Both blocks start at a height of 2 meters. Compare the kinetic energies (KE) of the blocks just before they reach the bottom, assuming there is no friction.
To compare the kinetic energies of blocks A and B, we need to use the formula for kinetic energy:
KE = (1/2) * m * v^2
where KE is the kinetic energy, m is the mass of the object, and v is its velocity.
Since the question states that both blocks are identical, we can assume that they have the same mass, so we can ignore the "m" factor in the equation.
To compare the kinetic energies, we need to calculate the velocities of the blocks just before they reach the bottom of the ramps. We can use basic trigonometry to calculate the velocities.
For Block A, the slope of the ramp is 45°, so we can calculate the velocity as follows:
vA = sqrt(2 * g * h)
where g is the acceleration due to gravity (approximately 9.8 m/s^2) and h is the initial height (2 meters). Substituting in these values, we get:
vA = sqrt(2 * 9.8 * 2) = sqrt(39.2) ≈ 6.26 m/s
For Block B, the slope of the ramp is 60°, so we can calculate the velocity as follows:
vB = sqrt(2 * g * h)
Using the same values of g and h, we get:
vB = sqrt(2 * 9.8 * 2) = sqrt(39.2) ≈ 6.26 m/s
Therefore, the velocities of both blocks just before they reach the bottom are the same: 6.26 m/s.
Now, we can compare their kinetic energies:
KEA = (1/2) * m * vA^2
KEB = (1/2) * m * vB^2
Since the mass is the same for both blocks, we can ignore the "m" factor and compare just the squares of the velocities:
KEA = (1/2) * vA^2
KEB = (1/2) * vB^2
Substituting the values of vA and vB, we get:
KEA = (1/2) * (6.26)^2 ≈ 19.54 J
KEB = (1/2) * (6.26)^2 ≈ 19.54 J
Therefore, the kinetic energies of blocks A and B just before they reach the bottom are equal with a value of approximately 19.54 Joules.