Sam, whose mass is 80kg , takes off across level snow on his jet-powered skis. The skis have a thrust of 220N and a coefficient of kinetic friction on snow of 0.1. Unfortunately, the skis run out of fuel after only 8.0s.
a) What is Sam's top speed?
b) How far has Sam traveled when he finally coasts to a stop?
a) Sam's top speed can be calculated using the equation v = (F/μm)t, where F is the thrust, μ is the coefficient of kinetic friction, m is the mass, and t is the time. Plugging in the given values, we get v = (220N/0.1*80kg) * 8.0s = 176 m/s.
b) The distance traveled can be calculated using the equation d = vt, where v is the velocity and t is the time. Plugging in the given values, we get d = 176 m/s * 8.0s = 1408 m.
To determine Sam's top speed and the distance he travels when he coasts to a stop, we need to use the concept of Newton's second law of motion and the equations of motion.
Let's start by calculating Sam's acceleration using Newton's second law of motion:
1. Calculate the net force acting on Sam:
Net force = Thrust - Frictional force
The thrust force is given as 220N, and the coefficient of kinetic friction on snow is 0.1. The frictional force can be calculated as:
Frictional force = coefficient of friction * weight
Weight = mass * gravitational acceleration (approximated as 9.8 m/s^2)
Frictional force = 0.1 * (80 kg * 9.8 m/s^2)
2. Calculate the net force:
Net force = 220N - Frictional force
3. Calculate Sam's acceleration:
Acceleration = Net force / mass
Next, we can calculate Sam's top speed using the equation of motion:
4. Use the equation:
Final velocity^2 = Initial velocity^2 + 2 * acceleration * distance
Sam's initial velocity is 0 m/s since he starts from rest.
Sam's final velocity is his top speed.
To find Sam's distance traveled, we need to know the time it takes for him to reach his top speed and the time it takes for him to come to a stop. Since we are given only the total time of 8.0 seconds, we will assume that Sam's acceleration is constant throughout this time period.
5. Calculate the distance traveled during the acceleration phase:
Distance = (Initial velocity * time) + (0.5 * acceleration * time^2)
6. Calculate the distance traveled during the coasting phase:
Distance = Top speed * (Total time - Acceleration time)
Let's calculate the values now:
a) To find Sam's top speed:
- Calculate the net force and the acceleration.
- Substitute the values in the equation:
Final velocity^2 = Initial velocity^2 + 2 * acceleration * distance.
- Take the square root of the final velocity to get the top speed.
b) To find Sam's distance traveled when he finally coasts to a stop:
- Calculate the distance traveled during the acceleration phase.
- Calculate the time it takes for Sam to reach his top speed.
- Calculate the distance traveled during the coasting phase.
I can perform these calculations if you provide the values for mass or any other missing information.
To find Sam's top speed, we can use the concept of net force and acceleration.
a) The net force acting on Sam can be calculated by subtracting the force due to friction from the thrust of the skis.
Net force = Thrust - Force of friction
The force of friction can be calculated using the formula:
Force of friction = coefficient of friction * normal force
The normal force in this case is equal to Sam's weight since he is on a level surface.
Normal force = mass * gravity
Given:
Mass (m) = 80 kg
Thrust = 220 N
Coefficient of friction (μ) = 0.1
Gravity (g) = 9.8 m/s^2
Substituting the values:
Normal force = m * g = 80 kg * 9.8 m/s^2 = 784 N
Force of friction = μ * Normal force = 0.1 * 784 N = 78.4 N
Net force = Thrust - Force of friction = 220 N - 78.4 N = 141.6 N
Now, using Newton's second law of motion, we can calculate the acceleration:
Net force = mass * acceleration
acceleration = Net force / mass = 141.6 N / 80 kg = 1.77 m/s^2
To find the top speed, we need to determine the distance traveled during the acceleration period. We can use the equation:
v^2 = u^2 + 2as
where v is final velocity, u is initial velocity (which is 0 m/s as Sam starts from rest), a is acceleration, and s is the distance traveled.
Solving for v:
v^2 = 0^2 + 2 * 1.77 m/s^2 * s
v^2 = 3.54 m/s^2 * s
v = √(3.54 m/s^2 * s)
Since Sam's top speed is reached when acceleration ceases (due to running out of fuel), we substitute the total time into the equation for v and solve for s:
8.0s = √(3.54 m/s^2 * s)
Square both sides to eliminate the square root:
(8.0s)^2 = (3.54 m/s^2 * s)
64s^2 = 3.54 m/s^2 * s
Divide both sides by 3.54 m/s^2:
64s = 1s
Solve for s:
s = (1s) / 64 = 0.0156s
Therefore, Sam's top speed is approximately 0.0156 m/s.
b) To find how far Sam traveled before he came to a stop, we need to calculate the distance covered during the acceleration phase and add it to the distance covered during the coasting phase.
The distance covered during acceleration can be calculated using the equation:
s = ut + (1/2)at^2
where u is the initial velocity, t is the time taken, a is the acceleration, and s is the distance covered.
Substituting the values:
s = 0 m/s * 8.0 s + (1/2) * 1.77 m/s^2 * (8.0 s)^2
s = 0 m + 7.08 m/s^2 * 64 s^2
s = 7.08 s * 64 s = 452.16 m
Therefore, Sam traveled approximately 452.16 meters before he came to a stop.