A 850.0-kg car travelling on a level road at 27.0 m/s (60.5 mi/hr) can stop, locking its wheels, in a distance of 61.0 m (200.1 ft).
(a) Find the size of the horizontal force which the car applies on the road while stopping.
(b) Find the stopping distance of that same car when it is traveling up a 18.1 deg slope, and it locks its wheels while traveling at 34.7 m/s (77.7 mi/hr). Assume that muk does not depend on the speed.
This is making no sense to me. Can I please have some help?
(a) (Force) x (stopping distance) = kinetic energy converted to heat
= (1/2) M V^2
Solve for F
(b) In this case, the initial kinetic energy is converted to both gravitational potential energy and heat.
The horizontal force will be whatever you get in Part (a), multiplied by cos 18.1 (0.9505). That is becasue the kinetic friction force is proportion to the normal component of the weight. Call this force F'
(Frictional work done) + (Potential energy increase) = (Initial kinetic energy)
F'*X + M g (X sin 18.1) = (1/2) M V^2
Solve for the new value of X.
The second term on the left is the increase in gravitational potential energy as the car sises up the slop.
I came up with 5079 N as my answer to a. Is that correct?
Yeppp n/m it is correct. Thanks a lot. :)
Of course, I'd be happy to help you!
To solve these problems, we can use Newton's laws of motion. Specifically, we'll use the second law, which states that the net force acting on an object is equal to its mass multiplied by its acceleration. In the first part of the problem, the car is stopping, so its acceleration is negative because it's slowing down.
(a) To find the size of the horizontal force applied by the car while stopping on a level road, we first need to find the car's acceleration. We can use the formula for acceleration:
a = (vf - vi) / t
Where:
a is the acceleration,
vf is the final velocity (0 m/s, since the car is stopping),
vi is the initial velocity (27.0 m/s),
and t is the time it takes for the car to stop.
We can rearrange the formula to solve for t:
t = (vf - vi) / a
Since we know vf, vi, and a, we can substitute those values into the formula to find t.
t = (0 - 27.0) / a
Next, we can use the formula for distance traveled during constant acceleration:
d = (vi * t) + (1/2) * a * t^2
Since we know d and t, we can solve for a:
61.0 = (27.0 * t) + (1/2) * a * t^2
Now we have two equations:
t = (0 - 27.0) / a
61.0 = (27.0 * t) + (1/2) * a * t^2
We can solve these equations simultaneously to find the acceleration, and then use that value to calculate the horizontal force applied by the car using Newton's second law.
(b) To find the stopping distance of the car when it is traveling up a slope, we need to consider the force of friction acting on the car. On a slope, there are two forces at play: the force of gravity pulling the car down the slope and the force of friction opposing the motion of the car.
The force of gravity acting down the slope can be calculated using the formula:
F_gravity = m * g * sin(θ)
Where:
m is the mass of the car (850.0 kg),
g is the acceleration due to gravity (9.8 m/s^2),
and θ is the angle of the slope (18.1 degrees).
The force of friction can be calculated using the formula:
F_friction = m * g * μk
Where:
μk is the coefficient of kinetic friction, which depends on the nature of the surfaces in contact.
Since in the problem, we are assuming that μk does not depend on the speed, we can use the same value for μk in both cases (horizontal and sloped).
To find the stopping distance on the slope, we need to equate the force applied by the car (F_applied) to the sum of the force of gravity and the force of friction:
F_applied = F_gravity + F_friction
Using Newton's second law, we can express these forces in terms of acceleration:
F_applied = m * a
F_gravity = m * g * sin(θ)
F_friction = m * g * μk
Since we know the values of m, g, θ, and μk, we can substitute those into the equations and solve for a. Then we can use the acceleration to find the stopping distance using the equation for distance traveled during constant acceleration.
I hope this explanation helps you understand how to approach the problem. Let me know if you have any further questions!