A car traveling at 90 km/h strikes a tree. The front end of the car compresses and the driver comes to rest after traveling 0.87 m. What was the average acceleration of the driver during the collision? Express the answer in terms of "g's," where 1.00g = 9.80 m/s2
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To find the average acceleration of the driver during the collision, we need to use the equation:
acceleration = (change in velocity) / (time taken)
First, let's find the change in velocity of the driver. Since the car comes to rest after the collision, the final velocity is 0. We need to convert the initial velocity from km/h to m/s, since the units of acceleration are m/s^2.
Given:
Initial velocity (v) = 90 km/h
Converting km/h to m/s: 1 km/h = 0.2778 m/s
Initial velocity (v) = 90 km/h * 0.2778 m/s = 24.9996 m/s (approximately 25 m/s)
Change in velocity = final velocity - initial velocity = 0 - 25 m/s = -25 m/s
Next, we need to find the time taken (t). We are given the distance traveled by the driver (0.87 m), but we need to find the time taken using the equation:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
Rearranging the equation, we have:
time^2 + (2 * initial velocity * time / acceleration) - (2 * distance / acceleration) = 0
This is a quadratic equation in terms of time, so we can solve it using the quadratic formula:
time = (-b ± sqrt(b^2 - 4ac)) / (2a)
Where a = 1, b = (2 * initial velocity / acceleration), and c = (-2 * distance / acceleration).
Plugging in the given values, we get:
time = (-2 * initial velocity ± sqrt((2 * initial velocity)^2 - 4 * 1 * (-2 * distance))) / (2 * 1)
time = (-2 * 25 ± sqrt(4 * 625 + 8 * distance)) / 2
time = (-50 ± sqrt(2500 + 8 * distance)) / 2
Simplifying, we have:
time = (-50 ± sqrt(2500 + 6.96)) / 2
time = (-50 ± sqrt(2506.96)) / 2
time ≈ (-50 ± 50.069) / 2
There are two possible solutions for time, but since the driver cannot travel back in time, we choose the positive square root:
time ≈ (-50 + 50.069) / 2 ≈ 0.0345 s (approximately)
Now that we have the change in velocity and the time taken, we can calculate the average acceleration:
acceleration = (change in velocity) / (time taken)
acceleration = (-25 m/s) / (0.0345 s) ≈ -724.64 m/s^2
Finally, we can express the answer in terms of "g's" by dividing the acceleration by the acceleration due to gravity:
acceleration in g's = (-724.64 m/s^2) / (9.80 m/s^2)
acceleration in g's ≈ -74 g's
Therefore, the average acceleration of the driver during the collision is approximately -74 g's.