In Fig, a runaway truck with failed brakes is moving downgrade at 160 km/h just before the driver steers the truck up a frictionless emergency escape ramp with an inclination of è = 11°. The truck's mass is 1.5 x 104 kg. (a) What minimum length L must the ramp have if the truck is to stop (momentarily) along it? (Assume the truck is a particle, and justify that assumption.) What is the minimum length L if (b) the truck's mass is decreased by 11% and (c) its speed is decreased by 11%?
The truck rises a distance of L sin 11, before momentarily stopping. At that point,
M g L sin 11 = (1/2) M Vo^2, where Vo is the initial velocity
Thus L = Vo^2/(2 g sin 11
For b, you will get the same answer, since M cancels out.
For (c), multiply the length required by (0.89)^2 = 0.792
Friction is being neglected. Often a gravel surface is used on esacape ramps to increase friction. The rotational kinetic energy of the wheels is also being neglected.
If The Angle is 11degree then we can use inclined plane equation
a=gsin0-f/m....1
by using 1st law of motion we have:
vf=vi+at.....2
To solve this problem, we need to consider the kinetic energy of the truck and the work done by the gravitational force while the truck is moving up the inclined ramp.
Let's start with part (a) and assume the truck is a particle. Justification for this assumption is that we are assuming the truck's mass is concentrated at a single point and we are neglecting any rotational effects. This approximation is valid when considering the overall motion and energy of the truck in this scenario.
(a) To find the minimum length L of the ramp for the truck to stop momentarily, we need to equate the work done by the gravitational force to the initial kinetic energy of the truck.
First, let's calculate the initial kinetic energy of the truck:
Kinetic energy (KE) = 0.5 * mass * velocity^2
KE = 0.5 * 1.5 x 10^4 kg * (160 km/h)^2
Since the units are not consistent, we need to convert the velocity from km/h to m/s:
160 km/h = 160 * (1000 m / 3600 s) = 44.44 m/s
Plugging the values into the equation:
KE = 0.5 * 1.5 x 10^4 kg * (44.44 m/s)^2
Now, let's calculate the work done by the gravitational force:
Work (W) = force * distance
The force acting on the truck is its weight, which can be calculated as:
Weight = mass * gravitational acceleration
Weight = 1.5 x 10^4 kg * 9.8 m/s^2
To calculate the distance, we can use the length of the ramp L and the inclination angle θ:
Distance = L * sin(θ)
Distance = L * sin(11°)
The work done by gravity is equal to the change in gravitational potential energy as the truck moves up the ramp:
W = m * g * Δh
Since the truck starts from rest at the bottom of the ramp and momentarily stops at the top, the change in height (Δh) is just the height of the ramp h:
Δh = h = L * cos(θ)
Now, we can equate the work done by gravity to the initial kinetic energy of the truck:
W = KE
m * g * h = 0.5 * m * v^2
Simplifying the equation:
g * h = 0.5 * v^2
9.8 m/s^2 * L * cos(11°) = 0.5 * (44.44 m/s)^2
Solving for L, we get:
L = (0.5 * (44.44 m/s)^2) / (9.8 m/s^2 * cos(11°))
Calculating the value, we can find the minimum length L.
Now let's move on to parts (b) and (c) where we need to consider the changes in mass and velocity.
(b) If the truck's mass is decreased by 11%, we can calculate the new mass as:
New mass = 0.89 * 1.5 x 10^4 kg
Using the same equation as in part (a), we can calculate the new minimum length L with the decreased mass.
(c) If the truck's speed is decreased by 11%, we can calculate the new velocity as:
New velocity = 0.89 * 44.44 m/s
Again, using the equation from part (a), we can calculate the new minimum length L with the decreased velocity.
By following these steps, you can find the minimum length L for each scenario.
To solve this problem, we can use the principles of physics, specifically Newton's laws of motion and the concept of work and energy.
First, let's address part (a) of the question - finding the minimum length L of the ramp required for the truck to stop.
Assuming the truck is a particle means that we can treat it as a point with its entire mass concentrated at one location. This assumption is justified because the dimensions of the truck are not given, and we are only interested in its motion along the ramp.
To find the minimum length L, we need to consider the forces acting on the truck on the ramp. The only force acting on the truck is its weight, which can be decomposed into two components: one parallel to the incline (mg sinθ) and the other perpendicular to the incline (mg cosθ).
The force parallel to the incline, mg sinθ, provides the necessary force to stop the truck by doing work against its initial kinetic energy. The work done on the truck can be calculated using the work-energy principle:
Work = Change in kinetic energy
Since the truck starts from a speed of 160 km/h (or 44.4 m/s) and comes to a stop, the change in kinetic energy is equal to the initial kinetic energy:
Work = (1/2)mv^2
Now, let's plug in the values we know: mass (m = 1.5 x 10^4 kg) and speed (v = 44.4 m/s):
Work = (1/2)(1.5 x 10^4 kg)(44.4 m/s)^2
Solving this equation will give us the work done by the force parallel to the incline.
Next, we need to find the distance over which this work is done, which is the length of the ramp (L). The work done by a force over a distance is given by the formula:
Work = Force x Distance
In this case, the force is mg sinθ and the distance is L:
Work = (mg sinθ)L
We can equate these two expressions for work and solve for L:
(1/2)(1.5 x 10^4 kg)(44.4 m/s)^2 = (mg sinθ)L
Here, we have everything except L, so we can solve for it:
L = [(1/2)(1.5 x 10^4 kg)(44.4 m/s)^2] / (mg sinθ)
Plugging in the values of mass (m = 1.5 x 10^4 kg), speed (v = 44.4 m/s), gravitational acceleration (g = 9.81 m/s^2), and the inclination angle (θ = 11°), we can calculate the minimum length L required for the truck to stop.
Now, let's move on to parts (b) and (c) of the question.
For part (b), where the truck's mass is decreased by 11%, we need to calculate the new mass and then repeat the steps above to find the new minimum length L.
To calculate the new mass, we subtract 11% of the original mass from the original mass:
New mass = 1.5 x 10^4 kg - 0.11(1.5 x 10^4 kg)
Once we have the new mass, we can repeat the calculations to find the new minimum length L.
For part (c), where the truck's speed is decreased by 11%, we can directly use the new speed to calculate the new minimum length L. We can either use the original mass or the mass found in part (b) as per the question's requirements.
By following these steps, you can find the answers to parts (a), (b), and (c) of the question.