Final answer:
The speed of the crate at the bottom of the 0.5 m high ramp, after accounting for gravitational potential energy and work done against friction, is approximately 2.545 m/s.
Step-by-step explanation:
Determining the Speed of the Crate at the Bottom of the Ramp
To determine the speed of the crate at the bottom of the ramp, we need to take into account both the energy input due to gravity and the energy loss due to friction. The gravitational potential energy (GPE) at the top of the ramp is converted into kinetic energy (KE) and work done against friction as the crate slides down.
Firstly, we calculate the GPE using the formula: GPE = mgh, where 'm' is the mass of the crate, 'g' is the acceleration due to gravity (9.81 m/s2), and 'h' is the height of the ramp. The GPE of the crate at the top is therefore 3 kg × 9.81 m/s2 × 0.5 m = 14.715 J.
Next, we calculate the work done against friction. Over the 1 m length of the ramp, the work done against the constant frictional force of 5 N is given by: Work done by friction = friction force × distance = 5 N × 1 m = 5 J.
Now, the net work done on the crate is the initial GPE minus the work done against friction, which is 14.715 J - 5 J = 9.715 J. This net work is converted entirely into KE of the crate since it starts from rest. KE is given by ½mv2 where 'v' is the final speed of the crate at the bottom of the ramp. We set the net work equal to the KE to solve for 'v'.
9.715 J = ½ × 3 kg × v2
v2 = 9.715 J / (1.5 kg)
v2 = 6.477 m2/s2
v = √6.477 m2/s2
v ≈ 2.545 m/s
Therefore, the speed of the crate at the bottom of the ramp is approximately 2.545 m/s.