A car traveling at 100Km/hr smashes into a concrete abutment. Show that the collision is the equivalent to driving off the top of a ten story building. Allow 4 meter per story and neglect air resistance. Next, find the equivalent vertical height to represent the effect of a two car head-on collision with each car traveling 100Km/hr.
To show that the collision in the first scenario is equivalent to driving off the top of a ten-story building, we need to calculate the vertical height from which the car would need to fall in order to have the same impact as the collision with the concrete abutment.
1. Determine the initial velocity of the car: The car is traveling at 100 km/hr. Convert this speed to meters per second (m/s). Since 1 km is equal to 1000 meters and 1 hour is equal to 3600 seconds, we can calculate the speed in m/s as follows:
100 km/hr * (1000 m/1 km) * (1 hr/3600 s) = 27.78 m/s
2. Calculate the kinetic energy of the car: The kinetic energy (KE) of an object is given by the equation KE = 1/2 * m * v^2, where m is the mass of the object and v is its velocity. Since the mass of the car is not provided and not required for this calculation, we can ignore it for now.
KE = 1/2 * v^2 = 1/2 * (27.78 m/s)^2 = 385.92 J (rounded to two decimal places)
3. Equate the kinetic energy to the potential energy: In a head-on collision with the abutment, the car's kinetic energy is converted into potential energy, given that there is no air resistance. We can use the formula for gravitational potential energy, which is PE = m * g * h, where m is mass, g is acceleration due to gravity, and h is the vertical height.
PE = 1/2 * v^2 = m * g * h
Since we are neglecting the mass of the car, we can simplify the equation as:
1/2 * v^2 = g * h
Rearrange the equation to solve for h:
h = (1/2 * v^2) / g
4. Calculate the equivalent vertical height: The acceleration due to gravity, g, is approximately 9.8 m/s^2. Insert the relevant values into the equation:
h = (1/2 * 27.78 m/s)^2 / 9.8 m/s^2 = 39.14 meters (rounded to two decimal places)
Since each story is equivalent to 4 meters, divide the height by 4 to find the number of stories:
39.14 meters / 4 meters = 9.79 stories (rounded to two decimal places)
Therefore, the collision is the equivalent of driving off the top of a ten-story building.
To find the equivalent vertical height for a two-car head-on collision, we can use a similar approach.
1. Determine the initial velocity of each car: Both cars are traveling at 100 km/hr. Convert this speed to meters per second (m/s) using the same method as before:
100 km/hr * (1000 m/1 km) * (1 hr/3600 s) = 27.78 m/s
2. Calculate the total kinetic energy: Since there are two cars involved in the collision, we need to calculate the total kinetic energy by doubling the value obtained in step 2 of the previous scenario:
Total KE = 2 * 385.92 J = 771.84 J (rounded to two decimal places)
3. Equate the kinetic energy to the potential energy: Following the same steps as before:
PE = 1/2 * v^2 = g * h
1/2 * v^2 = g * h
Rearranging the equation to solve for h:
h = (1/2 * v^2) / g
4. Calculate the equivalent vertical height: Insert the relevant values into the equation:
h = (1/2 * 27.78 m/s)^2 / 9.8 m/s^2 = 39.14 meters (rounded to two decimal places)
Divide the height by 4 to find the number of stories:
39.14 meters / 4 meters = 9.79 stories (rounded to two decimal places)
Therefore, the effect of a two-car head-on collision at 100 km/hr is equivalent to driving off the top of a ten-story building.
To understand the equivalence of the collision to driving off a ten-story building, we can use the concept of kinetic energy.
First, let's calculate the initial kinetic energy of the car before the collision:
Given:
Mass of the car (m) = unknown
Velocity of the car (v) = 100 km/hr = 100,000 m/3600 s ≈ 27.78 m/s
Kinetic energy (KE) = 0.5 * m * v^2
Now, let's calculate the initial kinetic energy using the given velocity:
KE = 0.5 * m * 27.78^2
KE = 0.5 * m * 771.24
KE ≈ 385.62 * m
To find the equivalent vertical height, we equate the initial kinetic energy to the potential energy at the top of a ten-story building.
Potential energy = m * g * h
Assuming the height of each story is 4 meters, the total height of a ten-story building is 10 * 4 = 40 meters.
Thus, we equate the kinetic energy to the potential energy:
385.62 * m = m * g * 40
Divide both sides by m:
385.62 = g * 40
To find g, the acceleration due to gravity, we can use the standard value of approximately 9.8 m/s^2.
Therefore, g ≈ 9.8.
Now, we solve for 385.62 = 9.8 * 40:
385.62 = 392
Since this equation is not satisfied, it implies that the collision is NOT equivalent to driving off a ten-story building. The car's initial kinetic energy is less than the potential energy at the top of a ten-story building.
Next, let's find the equivalent vertical height to represent the effect of a two-car head-on collision, assuming both cars are traveling at 100 km/hr (27.78 m/s).
For a head-on collision, the total kinetic energy is the sum of the kinetic energies of both cars:
Total initial kinetic energy (KE_total) = 0.5 * m1 * v^2 + 0.5 * m2 * v^2
Since both cars are traveling at the same velocity, we have:
KE_total = 0.5 * m1 * v^2 + 0.5 * m2 * v^2
KE_total = 0.5 * (m1 + m2) * v^2
Using the same process as before, we equate the initial kinetic energy to the potential energy:
0.5 * (m1 + m2) * v^2 = m * g * h
Assuming the same value for g (acceleration due to gravity) and using the height of a ten-story building (40 meters), we can solve for h:
0.5 * (m1 + m2) * v^2 = m * g * 40
Divide both sides by m:
0.5 * (m1 + m2) * v^2 = g * 40
Now, we solve for 0.5 * (m1 + m2) * v^2 = 9.8 * 40:
0.5 * (m1 + m2) * 771.24 = 392
Divide both sides by 771.24:
0.5 * (m1 + m2) = 392 / 771.24
Simplify:
0.5 * (m1 + m2) ≈ 0.508
This equation indicates that the equivalent vertical height to represent the effect of a two-car head-on collision is approximately 0.508 meters.