On October 9, 1992 a 12.3 kg meteorite struck a car in Peekskill, NY; leaving a dent 22 cm deep in the trunk. If the meteorite struck the car with a speed of 550 m/s, what was the magnitude of the acceleration (assuming it was constant)?
First, we need to determine the distance the meteorite traveled before coming to a stop, which is half of the depth of the dent. This is because the meteorite would have been decelerated to the point of stopping and then accelerated in the opposite direction due to the collision:
distance = depth / 2 = 22 cm / 2 = 11 cm = 0.11 m
Next, we need to use the following kinematic equation:
v^2 - u^2 = 2as
where v = final velocity = 0 m/s (since the meteorite came to a stop), u = initial velocity = 550 m/s, s = distance = 0.11 m, and a = acceleration.
Rearranging the formula to find acceleration:
a = (v^2 - u^2) / 2s
Plugging in the values:
a = (0^2 - 550^2) / (2 × 0.11)
a = (-302500) / 0.22
a = -1375000 m/s^2
Since acceleration is a vector, we are only interested in the magnitude, not the direction, the acceleration is just:
|a| = 1375000 m/s^2.
To find the magnitude of acceleration, we need to use the equation that relates the change in velocity to the acceleration and the time interval. In this case, since the velocity of the meteorite before and after the impact is not mentioned, we can assume that it comes to a stop after hitting the car.
The equation for the change in velocity can be rewritten as:
Change in velocity = Final velocity - Initial velocity
So, in this case, the initial velocity is 550 m/s (since the meteorite was moving before impact), and the final velocity is 0 m/s (since it comes to a stop after impact).
Let's assume that the time interval for the change in velocity is t seconds. Therefore, the change in velocity can be calculated as:
Change in velocity = Final velocity - Initial velocity = 0 m/s - 550 m/s = -550 m/s
Now, we need to determine the time it takes for the meteorite to come to a stop.
The equation that relates acceleration, change in velocity, and time is:
Change in velocity = acceleration * time
In this case, the change in velocity is -550 m/s (since the meteorite comes to a stop), and the time is the unknown parameter we are trying to find.
Rearranging the equation, we have:
Time = Change in velocity / Acceleration
Substituting the known values, we can solve for acceleration:
-550 m/s = acceleration * time
Acceleration = -550 m/s / time
To find the magnitude of acceleration, we need to find the time it takes for the meteorite to come to a stop. Unfortunately, the time is not directly given in the problem, so we need more information or assumptions to solve for acceleration.