Final answer:
The magnitude of the crate's acceleration depends on the components of gravitational force along the incline, while the frictional force can be found using Newton's second law by subtracting the net force from gravitational force parallel to the incline. The coefficient of kinetic friction is determined by dividing the frictional force by the normal force, and the final speed is calculated using a kinematic equation.
Step-by-step explanation:
To solve for the magnitude of the acceleration of the crate (part a), we can use the kinematic equation:
s = ut + \(rac{1}{2}\)at^2,
where s is the displacement (2.00 m), u is the initial velocity (0 m/s), t is the time (1.45 s), and a is the acceleration. Rearranging the equation to solve for a, we get:
a = \(\frac{2s}{t^2}\).
However, since the crate is on an incline, we need to consider only the component of gravitational acceleration parallel to the incline, which is g sin(\( heta\)), where g = 9.81 m/s^2 and \( heta\) is the angle of inclination (24.0°). This becomes more complex with the presence of friction, but as it is not factored into the kinematic equation directly, we’ll calculate it separately in part b.
To find the frictional force acting on the crate (part b), we can use Newton's second law:
\(F_{net} = ma\),
and the fact that the net force is the result of gravitational force parallel to the incline minus the frictional force. The gravitational force parallel to the incline can be calculated as:
\(F_{parallel} = mg\sin(\theta)\).
After finding the net force from part a and the parallel component of the gravitational force, the frictional force (F_friction) can be found by:
\(F_{net} = F_{parallel} - F_{friction}\).
Rearranging, we get:
\(F_{friction} = F_{parallel} - F_{net}\).
The next step is to calculate the coefficient of kinetic friction (part c) using the formula:
\(\mu_k = \frac{F_{friction}}{N}\),
where \(\mu_k\) is the coefficient of kinetic friction and N is the normal force. The normal force for an inclined plane is calculated as:
\(N = mg\cos(\theta)\).
Finally, to determine the speed of the crate after sliding 2.00 m (part d), we'll use the kinematic equation:
v = u + at,
where v is the final velocity, which we aim to find.