A 291.5 kg motorcycle is accelerating up along a ramp that is inclined 30.0° above the horizontal. The propulsion force pushing the motorcycle up the ramp is 3150 N, and air resistance produces a force of 250 N that opposes the motion. Find the magnitude of the motorcycle's acceleration.
We'll begin by resolving the forces parallel and perpendicular to the ramp.
The force of gravity acting on the motorcycle can be resolved into two components:
$F_g\sin\theta = (291.5\text{ kg})(9.81\text{ m/s}^2)\sin 30.0° \approx 1422\text{ N}$
$F_g\cos\theta = (291.5\text{ kg})(9.81\text{ m/s}^2)\cos 30.0° \approx 2537\text{ N}$
The propulsion force and air resistance force are already given as 3150 N and 250 N, respectively.
Resolving the forces parallel to the ramp:
$F_{parallel} = F_{propulsion} - F_{air\;resistance} - F_g\sin\theta$
$F_{parallel} = 3150\text{ N} - 250\text{ N} - 1422\text{ N} = 1478\text{ N}$
Resolving the forces perpendicular to the ramp:
$F_{perpendicular} = F_g\cos\theta$
$F_{perpendicular} = 2537\text{ N}$
Now we can use the equation for net force to find the acceleration:
$F_{net} = ma$
$F_{parallel} = ma$
$a = \frac{F_{parallel}}{m} = \frac{1478\text{ N}}{291.5\text{ kg}} \approx 5.07\text{ m/s}^2$
Therefore, the magnitude of the motorcycle's acceleration is approximately 5.07 m/s².