A car starts rolling down a 1-in-6 hill (1-in-6 means that for each 6 m traveled along the road, the elevation change is 1 m).
How fast is it going when it reaches the bottom after traveling 70 m ? Assume an effective coefficient of friction equal to 0.10
To determine the speed of the car when it reaches the bottom of the hill, we need to consider the physics involved. Here is a step-by-step explanation of how to solve this problem:
1. Start by calculating the vertical drop of the hill. Since the hill has a gradient of 1-in-6, it means that for every 6 meters traveled horizontally, there is a 1-meter change in elevation. Therefore, for a distance of 70 meters, the vertical drop would be (70 meters / 6) = 11.67 meters.
2. Next, calculate the net force acting on the car as it descends the hill. The two main forces at play here are gravity and friction. Gravity acts downward and is given by the equation: F_gravity = m*g, where m is the mass of the car and g is the acceleration due to gravity (approximately 9.8 m/s^2).
3. The force of friction opposes the motion of the car and can be calculated using the equation: F_friction = μ*N, where μ is the coefficient of friction and N is the normal force acting on the car. In this case, the normal force is equal to the gravitational force (F_gravity), since the car is on a downward slope.
4. Therefore, the net force acting on the car is given by the equation: F_net = F_gravity - F_friction.
5. Now, we can calculate the net force by substituting the values: F_net = m*g - μ*N.
6. To find the acceleration of the car, divide the net force by the mass of the car: a = F_net / m.
7. Since the car starts from rest, we can use the kinematic equation: v^2 = u^2 + 2*a*s, where v is the final velocity, u is the initial velocity (which is 0 in this case), a is the acceleration, and s is the distance traveled.
8. Rearranging the kinematic equation, we get: v^2 = 2*a*s.
9. Plug in the values and solve for v: v = sqrt(2*a*s).
10. Finally, substitute the calculated values and solve for v to get the velocity of the car at the bottom of the hill.
Note: The values of the car's mass and initial velocity are not given in the question. Therefore, you need those values to get an accurate answer.
To calculate the speed of the car when it reaches the bottom of the hill, we need to consider the force acting on the car.
1. First, let's calculate the change in elevation of the hill. According to the given information, for every 6 meters traveled, there is a 1-meter change in elevation. So, for 70 meters traveled, the change in elevation would be:
Change in Elevation = (70 m) / (6 m) = 11.67 m
2. Now, let's calculate the gravitational potential energy (GPE) lost by the car due to the change in elevation. The formula for GPE is given by:
GPE = m * g * h
Where:
m = mass of the car (assumed to be 1 kg for simplicity)
g = acceleration due to gravity (approximated to 9.8 m/s^2)
h = change in elevation
GPE = (1 kg) * (9.8 m/s^2) * (11.67 m)
3. Next, we need to consider the work done against friction. The formula for the work done is given by:
Work = force * distance
The force of friction can be calculated by multiplying the coefficient of friction with the normal force. The normal force can be calculated by multiplying the mass of the car with the acceleration due to gravity.
Force of Friction = coefficient of friction * (mass * acceleration due to gravity)
Work = (Force of Friction) * (distance)
In this case, distance = 70 m.
4. The change in kinetic energy of the car is equal to the work done against friction. The formula for kinetic energy is given by:
Kinetic Energy = 0.5 * m * v^2
Where:
m = mass of the car (assumed to be 1 kg for simplicity)
v = velocity of the car
The initial kinetic energy is assumed to be zero.
Change in Kinetic Energy = Kinetic Energy - Initial Kinetic Energy
= Kinetic Energy - 0.5 * m * 0^2
= Kinetic Energy
Therefore, the change in kinetic energy of the car is equal to the work done against friction.
5. Equating the work done against friction to the change in gravitational potential energy, we can solve for the velocity of the car when it reaches the bottom of the hill.
Change in Kinetic Energy = GPE
Substituting the values calculated in Steps 2 and 4:
Kinetic Energy = (1 kg) * (9.8 m/s^2) * (11.67 m)
Then:
(1/2) * (1 kg) * v^2 = (1 kg) * (9.8 m/s^2) * (11.67 m)
Simplifying:
v^2 = (2 * 9.8 m/s^2 * 11.67 m)
v = square root of (2 * 9.8 m/s^2 * 11.67 m)
Calculating the square root:
v ≈ 13.32 m/s
Therefore, the car will be traveling at approximately 13.32 m/s when it reaches the bottom of the hill after traveling 70 m.