Answer :
(a) Displacement is a vector quantity.
(b) The reasonable units for the quantity of kinetic energy multiplied by distance are kg·m²/s².
(c) The total energy of the block and spring system is 275.2 J.
(d) The block travels a distance of 0.75 m before coming to a stop.
The vector quantity among the options is (a) Displacement, as it has both magnitude and direction. Mass, time, and kinetic energy are scalar quantities as they only have magnitude, while speed is the magnitude of velocity and does not have direction.
The new physical quantity obtained by multiplying kinetic energy and distance has units of (b) kg·m²/s². This can be derived by multiplying the units of kinetic energy (kg·m²/s²) with the units of distance (m), resulting in kg·m²/s².
To calculate the total energy of the block and spring system, we use the formula for potential energy stored in a spring, which is given by E = (1/2)kx², where E is the energy, k is the spring constant, and x is the displacement. Plugging in the values, we get E = (1/2) * 80 N/m * (1.56 m)² = 97.3 J. Therefore, the total energy of the block and spring system is (e) 97.3 J.
For the block sliding on a rough horizontal surface, we need to calculate the work done against friction to bring the block to a stop. The work done against friction is equal to the initial kinetic energy of the block. Using the formula for kinetic energy (KE = 1/2 mv²), we find KE = 1/2 * 7 kg * (6.5 m/s)² = 147.875 J.
Since the work done against friction is equal to the initial kinetic energy, the distance traveled by the block is given by the work done against friction divided by the force of friction (W = Fd), which gives us d = KE / (μmg), where μ is the coefficient of friction and m is the mass of the block. Plugging in the values, we get d = 147.875 J / (0.46 * 7 kg * 9.8 m/s²) ≈ 0.75 m. Therefore, the block travels a distance of (d) 0.75 m before coming to a stop.
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