Question

A freight company uses a compressed spring to shoot $2.0kg$packages up a -high frictionless ramp into a truck, as FIGURE $P10.51$shows. The spring constant is $500N/m$and the spring is compressed $30cm$.

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

b. A careless worker spills his soda on the ramp. This creates a $50-cm$-long sticky spot with a coefficient of kinetic friction $0.30$. Will the next package make it into the truck?

Short answer

(a) The speed of the package when it reaches the truck is $1.7m/s$.

(b) The package will not make it to the truck.

Step-by-step solution

Given information (part a)

It is given that a freight company uses a compressed spring to shoot $2kg$packages up a $1m$high frictionless ramp into a truck.

The spring constant is given as $500N/m$and the spring is compressed $30cm$.

If soda is spilled on the ramp, it creates a $50cm$long sticky spot with a coefficient of kinetic friction $0.30.$

Explanation (part a)

Let v be the velocity of the package. When the spring is compressed, by the conservation of energy the kinetic energy transfers into the potential energy.

$$\begin{gathered} \frac{1}{2}m{v}^2=\frac{1}{2}k{x}^2-mgh \\ {v}^2=\frac{k{x}^2-2mgh}{m} \\ {v}^2=\frac{500{\left(0.30m\right)}^2-2\left(2Kg\right)\left(9.8m/se{c}^2\right)\left(1\right)}{2} \\ {v}^2=2.88m/sec \\ v=1.7m/sec \end{gathered}$$

The speed of the package when it reaches the truck is$1.7m/sec$

Given information (part b)

A freight company uses a compressed spring to shoot $2kg$packages up a $1m$high friction-less ramp into a truck. The spring constant is $500N/m$and the spring is compressed $30cm$. If soda is spilled on the ramp, it creates a $50cm$long sticky spot with a coefficient of kinetic friction

Explanation (part b)

The energy that is stored in the spring will be transferred to the package and will be used in doing work against the force of gravity and the friction force.

The work done in making the package onto the ramp is given as :

$$\begin{gathered} W=\text{Friction Force}\times \text{distance}+\text{Work done against gravity} \\ W=\left(\mu mg\right)d+mgh \\ W=m(\mu d+h) \\ W=2\times 9.81\left(0.3\times 0.5+1\right) \\ W=22.563 \end{gathered}$$

Since $W=22.563>\frac{1}{2}k{x}^2=22.5$, the package will not make it to the truck.