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.
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.
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