A stationary block explodes into two pieces L and R that slide across a frictionless floor and then into regions with friction, where they stop. Piece L, with a mass of 2.0 kg, encounters a coefficient of kinetic friction µL = 0.40 and slides to a stop in distance dL = 0.15 m. Piece R encounters a coefficient of kinetic friction µR = 0.50 and slides to a stop in distance dR = 0.20 m. What was the mass of the original block?
the momentum of L and R are equal.
energyL= mu*mg*distance
then solve for velocity L (KE=energy)
knowing velocityL, you can use momentum to find veloicty R.
I did:
energyL = mu*mg*distance
energyL = (.4)(2)(9.8)(.15)
energyL = 1.176
Then,
KE = 1/2mv^2
1.176 = 1/2(2)v^2
v = 1.08 m/s
Then which equation do I use???
To find the mass of the original block, you can use the principle of conservation of momentum. Since the momentum of L and R are equal after the explosion, you can set up an equation with the momentum of L equal to the momentum of R.
Momentum is given by the equation:
momentum = mass × velocity
Let's say the mass of the original block is M kg (which we need to find), and the velocity of L and R is v m/s.
For piece L: momentumL = massL × velocityL = 2.0 kg × v m/s
For piece R: momentumR = massR × velocityR
Since the momentum of L and R are equal:
2.0 kg × v m/s = massR × velocityR
Now, let's find the velocity of R. Since piece R stops after sliding a distance dR = 0.20 m, we can use the equation for energy (kinetic energy) to find the velocity:
energyR = frictional force × distance
The frictional force is given by the product of the coefficient of kinetic friction (µR), the mass of R, and the acceleration due to gravity (9.8 m/s^2), so:
energyR = µR × massR × 9.8 m/s^2 × dR = 0.50 × massR × 9.8 m/s^2 × 0.20 m
Now, we can equate kinetic energy (energyR) to the formula for kinetic energy:
energyR = 1/2 × massR × velocityR^2
0.50 × massR × 9.8 m/s^2 × 0.20 m = 1/2 × massR × velocityR^2
Simplifying:
massR × velocityR^2 = (0.50 × 9.8 m/s^2 × 0.20 m) / (1/2) = 0.50 × 9.8 m/s^2 × 0.20 m × 2
massR × velocityR^2 = 0.98 m^2/s^2
Now, substituting the expression for momentumR and the value of velocityR^2 into the equation from earlier:
2.0 kg × v m/s = massR × velocityR
2.0 kg × v m/s = 0.98 m^2/s^2
massR = (2.0 kg × v m/s) / (0.98 m^2/s^2)
massR = 2.04 kg/v
Now we know the mass of R in terms of v, the velocity of L. To solve for v, we can use the equation for kinetic energy:
KE = 1/2 × massL × velocityL^2 = energyL
Substituting the known values:
1.176 J = 0.5 × 2.0 kg × v^2
1.176 J = v^2
v = sqrt(1.176 J)
Now, substituting this value of v into the equation for massR:
massR = 2.04 kg / sqrt(1.176 J)
Finally, we can solve for the mass of the original block by adding the masses of L and R:
mass of original block = massL + massR
mass of original block = 2.0 kg + (2.04 kg / sqrt(1.176 J))
You can calculate the final answer using a calculator.