AN auto, moving too fast on a horizontal stretch of mountain road, slides off the road, falling into deep snow 43.9 meters below the road and 87.7 meters beyond the edge of the road.

a. How long did the auto take to fall?

b. How fast was it going when it left the road?

That is virtually the same as the problem I just did for you. Your turn.

To answer these questions, we need to use the principles of motion and the concepts of projectile motion.

a. To find the time it took for the auto to fall, we can use the equation of motion for vertical motion:

y = u*t + (1/2)*g*t^2

Where:
y = vertical distance (43.9 meters)
u = initial vertical velocity (unknown)
t = time taken (unknown)
g = acceleration due to gravity (approximately 9.8 m/s^2)

We can rearrange the equation to solve for time (t):

43.9 = u*t + (1/2)*9.8*t^2

This is a quadratic equation. To solve it, we can either factorize or use the quadratic formula. Let's use the quadratic formula:

t = (-b ± √(b^2 - 4*a*c)) / (2*a)

Where:
a = (1/2)*g = 4.9
b = u
c = -43.9

Substituting the values into the formula, we get:

t = (-u ± √(u^2 - 4*4.9*(-43.9))) / (2*4.9)

Simplifying the equation further, we get:

t = (-u ± √(u^2 + 856.12)) / 9.8

Since we know that the time taken cannot be negative, we can disregard the negative sign in front of the square root. So, the equation becomes:

t = (-u + √(u^2 + 856.12)) / 9.8

Now, we can plug in the values and solve for time (t).

b. To find the initial horizontal velocity (speed) of the auto when it left the road, we can use the equation of motion for horizontal motion. Since there is no horizontal force acting on the auto (assuming no air resistance), the initial horizontal velocity remains constant. So, the horizontal distance (87.7 meters) divided by the time taken (t) will give us the initial horizontal velocity (speed).

Velocity (speed) = Distance / Time

Speed = 87.7 / t

Now, we can substitute the value of t that we found in part (a) into this equation and solve for the speed.

Note: The speed we get will be the horizontal component of the auto's velocity when it left the road.