2. A person would like to jump over a river using a motorcycle and a ramp. The frictionless ramp is inclined at an angle of 42.5 degrees, has a length of 52 meters, and the motorcycle's engine produces a constant force of 915 Newton’s directed up the ramp, parallel to the ramp. The width of the river is 285 meters, and the landing on the other side of the river is at the same height at the end of the ramp. If the motorcycle is to just barely make it across the river and it's total mass is 66.4 kg, what is should the magnitude of the motorcycle’s velocity be at the bottom of the ramp in m/s?

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To find the magnitude of the motorcycle's velocity at the bottom of the ramp, we need to consider the forces acting on the motorcycle.

Step 1: Resolve the force produced by the motorcycle's engine into components.
The force produced by the engine can be resolved into two components: one parallel to the ramp and the other perpendicular to the ramp.
The force parallel to the ramp (F_par) is given by:
F_par = 915 Newton's

The force perpendicular to the ramp (F_perp) is given by:
F_perp = F_par * sin(42.5 degrees) (resolve the force using trigonometry)

Step 2: Determine the net force acting on the motorcycle.
The net force acting on the motorcycle is the sum of the parallel component of the engine's force and the force due to gravity.
The force due to gravity (F_gravity) is given by:
F_gravity = mass * gravitational acceleration
where the gravitational acceleration is approximately 9.8 m/s^2.

Step 3: Calculate the acceleration of the motorcycle.
The net force is given by:
net force = F_par + F_perp + F_gravity

Using Newton's second law of motion, we have:
net force = mass * acceleration

Therefore, we can solve for the acceleration:
acceleration = (F_par + F_perp + F_gravity) / mass

Step 4: Calculate the distance traveled along the ramp.
The distance traveled along the ramp can be calculated using the length of the ramp and the angle of inclination.

distance ramp = ramp length * cos(42.5 degrees)

Step 5: Calculate the magnitude of the motorcycle's velocity at the bottom of the ramp.
To calculate the magnitude of the motorcycle's velocity at the bottom of the ramp, we can use the kinematic equation:

vf^2 = vi^2 + 2 * acceleration * distance ramp

Since the motorcycle starts from rest at the top of the ramp, the initial velocity (vi) is 0.

Therefore, we can solve for the magnitude of the final velocity (vf):
vf = sqrt(2 * acceleration * distance ramp)

Now we can plug in the values and calculate the magnitude of the motorcycle's velocity.

Note: It is assumed that air resistance and other factors are negligible.

To find the magnitude of the motorcycle's velocity at the bottom of the ramp, you can use the principle of conservation of energy.

First, let's find the gravitational potential energy at the top of the ramp. The formula for gravitational potential energy is given by:

PE = m * g * h

where m is the mass (66.4 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height.

Since the height at the top of the ramp is given as the same height at the end of the ramp, the gravitational potential energy at the top is:

PE_top = m * g * h

Next, let's find the work done by the motorcycle's engine. The formula for work is given by:

W = F * d * cos(theta)

where F is the force (915 N), d is the displacement along the ramp (52 m), and theta is the angle between the force and the displacement (42.5 degrees).

Since the force and displacement are along the ramp, theta is 0 degrees, and cos(0) is equal to 1. So, the work done by the engine is:

W = F * d * cos(theta) = 915 N * 52 m * cos(0) = 47680 N m = 47680 J

Now, let's find the kinetic energy at the bottom of the ramp. The formula for kinetic energy is given by:

KE = (1/2) * m * v^2

where v is the velocity at the bottom of the ramp.

Since the motorcycle is to just barely make it across the river, the kinetic energy at the bottom of the ramp should be equal to the sum of the gravitational potential energy at the top and the work done by the engine:

KE_bottom = PE_top + W

Now, let's solve for the velocity v. Rearranging the kinetic energy formula, we get:

v = sqrt((2 * KE) / m)

Substituting the values into the equation:

v = sqrt((2 * (PE_top + W)) / m)

v = sqrt((2 * (m * g * h + W)) / m)

v = sqrt((2 * ((66.4 kg * 9.8 m/s^2 * h) + 47680 J)) / 66.4 kg)

Simplifying further:

v = sqrt((2 * (6472.96 h + 47680 J)) / 66.4 kg)

Now, plug in the given values for h = 0 (since it's the same height at the end of the ramp):

v = sqrt((2 * (6472.96 * 0 + 47680 J)) / 66.4 kg)

v = sqrt((2 * (0 + 47680 J)) / 66.4 kg)

Finally, calculate the magnitude of the motorcycle's velocity at the bottom of the ramp by evaluating the expression:

v = sqrt((2 * (47680 J)) / 66.4 kg)

v = sqrt(1437.35 m^2/s^2 / 66.4 kg)

v = sqrt(21.6511 m^2/s^2)

v ≈ 4.65 m/s

Therefore, the magnitude of the motorcycle's velocity at the bottom of the ramp is approximately 4.65 m/s.