A freight company uses a compressed spring to shoot 1.90kg packages up a 1.00-m

Question

A freight company uses a compressed spring to shoot 1.90kg packages up a 1.00-m -high frictionless ramp into a truck, as the figure(Figure 1) shows. The spring constant is 318N/m and the spring is compressed 35.0cm

What is the speed of the package when it reaches the truck?

A careless worker spills his soda on the ramp. This creates a 50.0-cm-long sticky spot with a coefficient of kinetic friction 0.300. Will the next package make it into the truck?

A freight company uses a compressed spring to shoot 1.90kg packages up a 1.00-m -high frictionless ramp into a truck, as the figure(Figure 1) shows. The spring constant is 318N/m and the spring is compressed 35.0cm What is the speed of the package when it reaches the truck? A careless worker spills his soda on the ramp. This creates a 50.0-cm-long sticky spot with a coefficient of kinetic friction 0.300. Will the next package make it into the truck?

Explanation / Answer

Case 1 ( frictionless)
Conserving the energy .
so,

1/2 * k x^2 = mgh + 1/2 mv^2

so,

1/2 * 318* ( 0.35^2) = 1.9 * 9.81 * 1 + 1/2 * 1.9 * v^2

solving it ,

v= 0.9395 m/s ~ 0.94 m/s

This is the speed of the package when it reaches the truck.

Case 2 : ( with friction)
In this case, some energy is used by friction as negative work done by friction
So ,
Friction force = mu* N

Friction Force = 0.3* 1.9* 9.81 = 5.5917 N

so,
energy used by friction = -ve of Work done by friction

energy used by friction = F * d = 5.5917 * 0.5 = 2.79585 J
so,
Again
Conserving the energy .
so,

1/2 * k x^2 = mgh + 1/2 mv^2 + energy used by friction

so,

1/2 * 318* ( 0.35^2) = 1.9 * 9.81 * 1 + 1/2 * 1.9 * v^2 + 2.79585

solving it ,

v comes out to be imaginary .

so , therefore , Block will not reach the top of the truck in this case.

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