In a front-end collision, a 1400 kg car that has shock-absorbing bumpers can only withstand a maximum force of 75 kN before any sort of damage occurs.If the maximum speed for a car that is nondamaged in a collision is 10 Km/h, by how much must the bumper be able to move relative to the car?
I got 0.068 meters, where did i go wrong?
The work done as the bumper is compressed the maximum distance with no damage, which is Fmax*X/2, must exceed the kinetic energy of the car. (The factor of 1/2 comes in because the force increases linearly with compression.)
X = [2/Fmax)]*(1/2)*M*Vmax^2
X = M*Vmax^2/Fmax
Convert 10 km/h to ? m/s before using the formula.
I get 0.144 meters. Your error may be in the factor of 1/2
Unless specified, linear compression probably has no bearing on the problem.
.5mV^2 = F*D is the correct formula.
10 kpm = 2.77 m/s
0.072 is the best answer.
Well, it seems like you've done some math there, but let me check your answer. To calculate by how much the bumper must be able to move relative to the car, we need to relate the maximum force and the maximum speed of the car.
Let's convert the car's maximum speed to meters per second. We have 10 km/h, which is approximately 2.78 m/s.
Now, we can use Newton's second law, which states that force equals mass times acceleration. The acceleration here is the change in velocity over time. Since the car is stopping, we'll consider it a negative acceleration.
We can use the kinematic 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 displacement.
In this case, the final velocity is 0 m/s (as the car stops), the initial velocity is 2.78 m/s (as given), and the acceleration is what we are trying to find. The displacement (the movement of the bumper) is also what we want to determine.
Rearranging the equation, we have s = (v^2 - u^2) / (2a).
Plugging in the values, we get s = (0 - 2.78^2) / (2a), which simplifies to s = -2.78^2 / (2a).
Now let's solve for a:
a = -2.78^2 / (2s).
We know that the bumper can withstand a maximum force of 75 kN, which we can convert to Newtons by multiplying by 1000. So we have 75 kN * 1000 = 75,000 N.
The maximum force can be calculated as F = ma. Rearranging the equation, we have a = F / m.
Plugging in the values, we get a = 75,000 N / 1400 kg, which simplifies to a = 53.57 m/s^2.
Now we can calculate the displacement:
s = -2.78^2 / (2 * 53.57), which gives us s ≈ -0.07 m.
So, it seems like you were close! It looks like you made a rounding error, which led to a slightly different answer. The correct value should be around -0.07 m, or approximately -7 cm (since we take the negative sign into account as the car is decelerating).
To find the correct answer, you need to use the equation of motion and energy conservation principles. Let's go through the steps to calculate the correct displacement.
Step 1: Convert the given maximum speed from Km/h to m/s.
Given maximum speed = 10 Km/h
Converting to m/s:
1 km = 1000 m
1 h = 3600 s
10 Km/h = 10 * 1000 / 3600 = 2.778 m/s
Step 2: Use the equation of motion to find the acceleration.
Initial velocity, u = 0 (car is at rest)
Final velocity, v = 2.778 m/s
Time, t = ?
Using the equation v = u + at, we can rearrange it to solve for acceleration:
a = (v - u) / t
a = (2.778 - 0) / t
a = 2.778 / t
Step 3: Use Newton's second law to find the force generated during the collision.
Mass, m = 1400 kg
Force, F = ?
Acceleration, a = from Step 2
Using the equation F = ma, we can substitute the values:
F = 1400 * a
Step 4: Determine if the force generated during the collision exceeds the maximum force the car can withstand.
Maximum force = 75 kN = 75,000 N
If F > Maximum force, there will be damage to the car.
Step 5: Calculate the displacement of the bumper using energy conservation principles.
Displacement, s = ?
Force, F = from Step 3
Maximum force = 75,000 N
Work done, W = F * s (Assuming the force is constant)
Kinetic energy, KE = (1/2) * m * v^2 (Assuming no energy is lost in the collision)
Since work is equal to the change in kinetic energy (W = KE), we can set up the equation:
F * s = (1/2) * m * v^2
Substituting the values, we get:
75,000 * s = (1/2) * 1400 * (2.778)^2
75,000 * s = 5799
Step 6: Solve for the displacement, s.
s = 5799 / 75,000
s = 0.0773 m (rounded to four decimal places)
So, the correct displacement of the bumper relative to the car is approximately 0.0773 meters.
To determine the correct answer, we need to use the principles of physics. Let's break down the problem step by step:
1. First, convert the car's maximum speed from km/h to m/s. Since 1 km/h is equal to 0.2778 m/s, we can calculate as follows:
Maximum speed = 10 km/h * (0.2778 m/s / 1 km/h) = 2.778 m/s
2. Next, we need to calculate the maximum kinetic energy of the car before the collision. The kinetic energy of an object is given by the formula:
Kinetic energy = (1/2) * mass * velocity^2
Plugging in the values:
Kinetic energy = (1/2) * 1400 kg * (2.778 m/s)^2 = 5436.42 J (Joules)
3. Now, we'll calculate the work done by the force (work-energy principle). The work done is equal to the change in kinetic energy. In this case, the force applied is equal to the maximum force the car can withstand before being damaged (75 kN).
Work done = Force * distance
Rearranging the formula to solve for distance:
Distance = Work done / Force
Plugging in the values:
Distance = 5436.42 J / 75,000 N = 0.072 m (meters)
Therefore, the bumper must be able to move relative to the car by approximately 0.072 meters, not 0.068 meters as you calculated.