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.
I'm never sure which equation to use for these distance problems...
When dealing with distance problems involving friction, it is helpful to remember the equation for the net force acting on an object on an inclined plane:
Net force = mass * acceleration
In this case, the net force on the bottle cap is the force of gravity acting downwards minus the force of friction acting upwards. The force of friction can be calculated using the equation:
Force of friction = coefficient of friction * normal force
The normal force is the perpendicular force exerted by the inclined board on the bottle cap and can be found using trigonometry:
Normal force = mass * gravity * cos(θ)
Where θ is the angle of inclination.
The net force can also be expressed as:
Net force = mass * acceleration
Given that the cap is slowing down, the acceleration will be negative. Rearranging the equation, we have:
Acceleration = (final velocity - initial velocity) / time
Since the cap is slowing down, the acceleration will be negative.
Now, we can substitute the expressions for the net force and acceleration into the original equation:
mass * acceleration = mass * gravity * sin(θ) - (coefficient of friction) * mass * gravity * cos(θ)
Simplifying further, we get:
acceleration = gravity * (sin(θ) - coefficient of friction * cos(θ))
Finally, we can substitute the given values into the equation and solve for the distance:
d = (v_f^2 - v_i^2) / (2 * |acceleration|)
where d is the distance, v_f is the final velocity, v_i is the initial velocity, and |acceleration| is the magnitude of the acceleration.