Final answer:
Calculating the work done by the slingshot and equating it to the kinetic energy provides the rock's launch speed, which is 39.87 m/s. The maximum height, found using the conservation of energy, is 80.62 m, which does not match any of the provided options.
Step-by-step explanation:
To find the speed of the rock as it leaves the slingshot, we first calculate the work done by the slingshot using work-energy principle:
Work = Force × Distance = 53 N × 0.750 m = 39.75 J
Work done by slingshot is converted entirely into the rock's kinetic energy since air resistance is neglected:
Kinetic Energy = ½ m v2
Substituting the work done for kinetic energy, we can solve for the speed (v):
39.75 J = ½ × 0.050 kg × v2 => v = √(39.75 J / 0.025 kg) = √(1590) ≈ 39.87 m/s
The maximum height reached by the rock can be found using the conservation of energy, where all kinetic energy at the start becomes potential energy at the highest point:
Potential Energy at max height = Kinetic Energy at beginning
mgh = ½ m v2
Assuming g = 9.8 m/s2, and rearranging for h:
h = v2 / (2g) = (39.87 m/s)2 / (2 × 9.8 m/s2) ≈ 80.62 m
From these calculations, it seems none of the options provided (a, b, c, d) in the question match the correct calculated values.