A youngster shoots a bottle cap up a 15.0° inclined board at 1.92 m/s. The cap slides in a straight line, slowing to 0.95 m/s after traveling some distance. If the coefficient of kinetic friction is 0.35, find that distance.
Can someone please show me the steps to solving this one?
assume the bottle cap mass is m, it will not matter what m is.
There are only two forces on the mass in the direction down the incline, that due to the weight and that due to friction.
the amount due to the weight is:
m g sin 15 down the ramp parallel with the ramp
the amount due to friction is also down the ramp while the bottle cap is moving up
m g (.35) cos 15
so the total force down the ramp is:
m g ( sin 15 + .35)
that force is the mass times the acceleration down the ramp
m g (sin 15 + 3.5) = m ( acceleration down the ramp)
notice that we can divide both sides by m so the mass did not matter
so
a = 9.8 ( sin 15 + .35) DOWN the ramp
FIND that number and use it for a
if x is distance UP the ramp then
x = xo + vo t - .5 a t^2
v = vo - a t
vo = 1.92
vf = .95 = 1.92 - a t
solve that for t, the time during which it slows down
then use that t in
x = 0 + 1.92 t - .5 a t^2
I come up with 0.233 m as my answer, but that's not right. What am I doing wrong?
looks pretty good to me
I have no problem with your answer.
Hey wait a minute, I called cos 15 one. It should be .966
That will change the answer a little.
So wait...do I use cos 15 to find the acceleration?
Yes
But I thought acceleration was a = 9.8 ( sin 15 + .35)...
Sure! In order to find the distance traveled by the bottle cap, we need to use the equations of motion and the concept of kinetic friction.
Step 1: Draw a diagram and identify the given information:
- The initial velocity (u) of the bottle cap is 1.92 m/s.
- The final velocity (v) of the bottle cap is 0.95 m/s.
- The angle of the inclined board (θ) is 15.0°.
- The coefficient of kinetic friction (μ) is 0.35.
Step 2: Break down the initial velocity into its components:
The initial velocity of the bottle cap can be split into two components: the horizontal component (u_x) and the vertical component (u_y). Since the cap slides in a straight line, the vertical velocity remains constant, but the horizontal velocity decreases due to the effect of friction.
u_x = u * cos(θ)
u_x = 1.92 m/s * cos(15°)
u_y = u * sin(θ)
u_y = 1.92 m/s * sin(15°)
Step 3: Determine the acceleration along the incline:
The acceleration along the incline (a) is caused by the component of gravity acting parallel to the incline, which is given by a = g * sin(θ). Here, g represents the acceleration due to gravity.
a = 9.8 m/s^2 * sin(15°)
Step 4: Find the deceleration due to friction:
The deceleration (decrease in velocity) caused by kinetic friction can be calculated using the equation:
a_friction = -μ * g * cos(θ)
a_friction = -0.35 * 9.8 m/s^2 * cos(15°)
Step 5: Calculate the distance traveled:
Using the equation of motion v^2 = u^2 + 2as, we can substitute the known values to find the distance traveled.
0.95^2 = (1.92 * cos(15°))^2 + 2 * a_friction * s
Solve the equation for s (the distance traveled).
Step 6: Substitute the values and solve for s:
0.95^2 = (1.92 * cos(15°))^2 + 2 * (-0.35 * 9.8 * cos(15°)) * s
Now, solve for s to find the distance traveled by the bottle cap.