A 75kg snowboarder has an initial velocity of 5.0 m/s at the top of a 28 degree incline. After sliding down the 110m long incline (on which the coefficient of kinetic friction is = Bk0.18), the snowboarder has attained a velocity v . The snowboarder then slides along a flat surface (on which Bk= 0.15) and comes to rest after a distance x .
Part a) Use Newton's second law to find the snowboarder's acceleration while on the incline and while on the flat surface.
part b)Then use these accelerations to determine x .
m =75 kg, vₒ=5 m/s, s = 110 m,
k1 = 0.18, k2 = 0.15.
a1=? a2=?, x =?
The equations of the motion along the incline are
x: m•a1 = m•g•sinα – F1(fr),
y: 0 = - m•g•cosα + N.
F1(fr) = k1•N=k1•m•g•cos α,
m•a1 = m•g•sinα – k1•m•g•cos α,
a1 = g• (sinα - k1•cos α) = 9.8•(0.469 – 0.18•0.883) = 3.04 m/s².
v=vₒ +a1•t,
s =vₒ•t + a1•t²/2 .
v = sqrt(2•a1•s +vₒ²) = sqrt(2•3.04•110 + 25) = 26.33 m/s.
The motion along a flat surface:
m•a2 = F2(fr) = k2•m•g,
a2 = k2•g = 0.15•9.8 = 1.47 m/s².
x = v²/2•a2 = (26.33)²/2•1.47 = 235 m.
a=3.05
Part a)
On the incline:
Let's start by finding the gravitational force acting on the snowboarder while on the incline. The formula for gravitational force is F = m * g, where m is the mass (75 kg) and g is the acceleration due to gravity (9.8 m/s^2). Therefore, the gravitational force is F = 75 * 9.8 = 735 N.
Next, we'll find the force of friction on the incline. The formula for the force of friction is Ff = Bk * Fn, where Bk is the coefficient of kinetic friction (0.18) and Fn is the normal force. The normal force is equal to the gravitational force acting perpendicular to the incline, which can be calculated using the formula Fn = m * g * cos(theta), where theta is the incline angle (28 degrees). Therefore, Fn = 75 * 9.8 * cos(28) = 661.54 N.
Now, we can find the force of friction on the incline: Ff = 0.18 * 661.54 = 119.28 N.
Since the snowboarder is sliding down the incline, the force of friction acts in the opposite direction to the motion, so we can say Ff = -119.28 N.
Using Newton's second law, F = m * a, we can find the acceleration on the incline. The net force acting on the snowboarder while on the incline is the sum of the gravitational force and the force of friction, so Fnet = F + Ff. Therefore, Fnet = 735 + (-119.28) = 615.72 N.
Plugging this into Newton's second law and solving for acceleration: 615.72 = 75 * a. Therefore, a = 615.72 / 75 = 8.2096 m/s^2.
On the flat surface:
The gravitational force and normal force acting on the snowboarder are the same as before. Therefore, Fn = 661.54 N.
Using the same formula, the force of friction on the flat surface is calculated: Ff = 0.15 * 661.54 = 99.23 N.
Similarly, since the snowboarder is coming to rest on the flat surface, the force of friction acts in the opposite direction to the motion, so Ff = -99.23 N.
Again, using Newton's second law, F = m * a, we can find the acceleration on the flat surface. The net force acting on the snowboarder while on the flat surface is the sum of the gravitational force and the force of friction, so Fnet = F + Ff. Thus, Fnet = 735 + (-99.23) = 635.77 N.
Plugging this into Newton's second law and solving for acceleration: 635.77 = 75 * a. Therefore, a = 635.77 / 75 = 8.4769 m/s^2.
Part b)
To find the distance x, we need to find the time it takes for the snowboarder to come to rest on the flat surface. Since we know the acceleration on the flat surface is 8.4769 m/s^2, and the final velocity is 0 (since the snowboarder comes to rest), we can use the kinematic equation v^2 = u^2 + 2ad (where u is the initial velocity, v is the final velocity, a is the acceleration, and d is the distance) to solve for distance.
In this case, since u = v = 0 and a = 8.4769, the equation becomes 0 = 0 + 2 * 8.4769 * x. Simplifying, we find 16.9538x = 0, so x = 0.
Therefore, the snowboarder comes to rest on the flat surface immediately after sliding down the incline, so x = 0.
Part a) To find the snowboarder's acceleration while on the incline, we can use Newton's second law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration:
Net force = mass * acceleration
On the incline, the net force acting on the snowboarder is the force of gravity acting down the incline, minus the force of friction acting up the incline.
Force of gravity = mass * gravitational acceleration
Force of friction on the incline = coefficient of kinetic friction * normal force
Normal force on the incline = mass * gravitational acceleration * cos(angle of incline)
Now, we can substitute these equations into the equation for net force:
Net force = (mass * gravitational acceleration) - (coefficient of kinetic friction * mass * gravitational acceleration * cos(angle of incline)) = mass * acceleration
Simplifying, we get:
Acceleration on the incline = gravitational acceleration - (coefficient of kinetic friction * gravitational acceleration * cos(angle of incline))
Using the given values, the gravitational acceleration is approximately 9.8 m/s^2, the coefficient of kinetic friction on the incline is 0.18, and the angle of incline is 28 degrees.
Acceleration on the incline = 9.8 m/s^2 - (0.18 * 9.8 m/s^2 * cos(28 degrees))
Calculate this expression to find the snowboarder's acceleration while on the incline.
To find the snowboarder's acceleration while on the flat surface, we can use the same equation as above, but with the coefficient of kinetic friction on the flat surface and the normal force on the flat surface (which is equal to the snowboarder's weight since there is no incline).
Acceleration on the flat surface = gravitational acceleration - (coefficient of kinetic friction * gravitational acceleration * cos(0 degrees))
Using the given value for the coefficient of kinetic friction on the flat surface (0.15), calculate this expression to find the snowboarder's acceleration while on the flat surface.
Part b) To determine x, we need to use the accelerations calculated in part a. We can use the equations of motion to find the distance x when the snowboarder comes to rest:
Final velocity = 0
Initial velocity = velocity attained after sliding down the incline
Acceleration = acceleration on the flat surface
Using the equation v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the distance, we can rearrange it to solve for s:
s = (v^2 - u^2) / (2 * a)
Substituting the values, calculate the distance x by plugging in the final velocity as 0, the initial velocity as the velocity attained after sliding down the incline, and the acceleration as the acceleration on the flat surface.