Answer to Question #163410 in Physics for Johnny Tim

Question

A crossbow applies an average force F to an arrow for a time Δt1 and the arrow leaves the crossbow with a speed v. The arrow with mass m1 strikes and becomes embedded in a wooden block with mass m2, placed on a horizontal surface just in front of the crossbow. During the collision between the arrow and block, both experience an Impulse Δp. The coefficient of friction between the block and the surface is 𝛍k and the impact drives the block a distance d. 

1. Show that the distance d the crate slides is:      

1. Calculate the distance d that the crate slides before coming to a stop.                

Data: 

F = 667 N

Δt1 = 0.0038 s

m1 = 0.023 kg

m2 = 1.4 kg

𝛍k = 0.63

Answer 1

$$F\Delta t_1=m_1v$$

$$m_1v=(m_1+m_2)u=F\Delta t_1$$

$$0.5(m_1+m_2)u^2=\mu (m_1+m_2)gd$$

$$d=\frac{F^2\Delta t_1^2}{2\mu g (m_1+m_2)^2}$$

$$d=\frac{(667\cdot0.0038)^2}{2(0.63)(9.8)(0.023+1.4)^2}=0.27\ m$$