A box with an initial speed of #7 m/s# is moving up a ramp. The ramp has a kinetic friction coefficient of #3/2 # and an incline of #pi /4 #. How far along the ramp will the box go?

Question

A box with an initial speed of #7  m/s# is moving up a ramp.  The ramp has a kinetic friction coefficient of #3/2 # and an incline of #pi /4 #.  How far along the ramp will the box go?

Julia Aguilar · Accepted Answer

#sqrt2 #meters

We know that the initial velocity ( #v_i#) is #7ms^-1# and the box will move on the ramp which makes an angle #pi/4# with the horizontal till the velocity becomes zero . Sp that the final velocity ( #v_f# ) of the box is zero. The initial, final, and displacement (S) are connected by a kinematic relation that is given by #v_f^2-v_i^2=2*a*S# where The box's acceleration is represented by a. Thus, the box's acceleration is the only thing we need to know. Acceleration is defined as the force applied to the box in its direction of motion divided by its mass. #a=F/M# where M is the box's mass and F is the force. ONLY OPPOSING FORCES ARE AT WORK IN THE BOX. They are: 1. gravitational force; and 2. frictional force. Not the complete gravitational force acting on the box along it's motion only the negative of the vertical component acts along the direction of the motion that is #-Mg*sin(pi/4)# The frictional force #-mu Mgcos(pi/4)# in order for the net force acting on it to be total of the forces of friction and gravity #F=-(Mg)/sqrt2(1+mu)# where #mu# is the coefficient of kinetic friction .. Therefore, acceleration is #a=(cancel(M)g(1+mu))/(cancelMsqrt2)# INTERCHANGE THE VALUES TO SPEED UP: #a=(9.8*(1+3/2))/sqrt2=>(5xx9.8)/(2sqrt2)# Returning to the displacement relation #v_i=7 ms^-1# #v_f=0ms^-1# #7^2-0=2*a*s# #49/(2a)=S# #S=(49xx2sqrt2)/(98)=>sqrt2 m#

Charles Aeschliman · Answer

undefined To find the distance along the ramp, use the kinematic equation:

$$ d = \frac{{v_0^2 \sin(2\theta)}}{{g \cdot \mu_k + g \cdot \sin(\theta)}} $$

where:
- $ v_0 $ is the initial speed (7 m/s),
- $ \theta $ is the incline angle ($ \frac{\pi}{4} $),
- $ g $ is the acceleration due to gravity (9.8 m/s²),
- $ \mu_k $ is the kinetic friction coefficient (3/2).

Substitute the values to find $ d $.