The figure shows two packages that start sliding down a 20° ramp from rest a distance along the ramp from the bottom.Package A has a mass of 5.0 kg and a coefficient of kinetic friction 0.20 between it and the ramp.Package B has a mass of 10 kg and a coefficient of kinetic friction 0.15 between it and the ramp.How long does it take package A to reach the bottom?
A) 2.7 s
B) 3.0 s
C) 3.2 s
D) 3.5 s

Question

The figure shows two packages that start sliding down a 20° ramp from rest a distance  along the ramp from the bottom.Package A has a mass of 5.0 kg and a coefficient of kinetic friction 0.20 between it and the ramp.Package B has a mass of 10 kg and a coefficient of kinetic friction 0.15 between it and the ramp.How long does it take package A to reach the bottom? 
A) 2.7 s 
B) 3.0 s 
C) 3.2 s 
D) 3.5 s

Quizplus · Accepted Answer

First, we need to calculate the gravitational force acting on each package down the ramp. Using the formula F=mg(sinθ), where θ is the angle of the ramp (20°) and g is the acceleration due to gravity (9.8 m/s^2), we get:
- For package A: F=5.0 kg x 9.8 m/s^2 x sin(20°) = 16.9 N
- For package B: F=10 kg x 9.8 m/s^2 x sin(20°) = 33.8 N

Next, we need to calculate the force of kinetic friction acting on each package as it slides down the ramp. Using the formula f=μkN, where μk is the coefficient of kinetic friction and N is the normal force (equal to the component of the gravitational force perpendicular to the ramp), we get:
- For package A: f=0.20 x 5.0 kg x 9.8 m/s^2 x cos(20°) = 8.8 N
- For package B: f=0.15 x 10 kg x 9.8 m/s^2 x cos(20°) = 8.3 N

The net force acting on each package down the ramp is therefore:
- For package A: Fnet = F - f = 16.9 N - 8.8 N = 8.1 N
- For package B: Fnet = F - f = 33.8 N - 8.3 N = 25.5 N

We can now use the formula F=ma, where a is the acceleration of each package down the ramp, to solve for a:
- For package A: 8.1 N = 5.0 kg x a, so a = 1.62 m/s^2
- For package B: 25.5 N = 10 kg x a, so a = 2.55 m/s^2

Finally, we can use the formula x = 1/2at^2 + vt, where x is the distance traveled down the ramp (11 m), v is the initial velocity (0 m/s), and t is the time taken to travel the distance x, to solve for t:
- For package A: 11 m = 1/2 x 1.62 m/s^2 x t^2, so t = 2.68 s ≈ 2.7 s
- For package B: 11 m = 1/2 x 2.55 m/s^2 x t^2, so t = 2.79 s (this is not one of the answer choices, so we can eliminate B, C, and D)

The Figure Shows Two Packages That Start Sliding Down a 20°