An arrow is shot verticaaly upward. Two seconds later, it is at a height of 42 meters ( although this is not necessarily the highest point). with what speed was the arrow shot? how long is the arrow in flight altogether?

1. h = Vo*t + 0.5g*t^2 = 42m,

Vo*2 + (-4.9)*2^2 = 42,
2Vo - 19.6 = 42,
2Vo = 42 + 19.6 = 61.6,
Vo = 30.8m/s.

2. t(up) = (Vf - Vo / g,
t(up) = (0 - 30.8) / -9.8 = 3.14s.

t(dn) = t(up) = 3.14s.

T=t(up) + t(dn) = 3.14 + 3.14 = 6.28s.=
Time in flight.

To determine the initial velocity at which the arrow was shot vertically upward, we can use the concept of projectile motion.

First, let's define the motion of the arrow. When the arrow reaches its highest point, its vertical velocity will be zero. Therefore, we can determine the time it takes for the arrow to reach its highest point by using the equation:

vf = vi + gt

Where:
- vf is the final velocity at the highest point (which is zero because it momentarily stops),
- vi is the initial velocity (what we want to find),
- g is the acceleration due to gravity (approximately -9.8 m/s^2),
- t is the time it takes to reach the highest point.

Since we know that at t = 2 seconds the arrow is at a height of 42 meters, we can use the equation for displacement in vertical motion:

d = vit + 0.5gt^2

Where:
- d is the displacement (42 meters in this case),
- vi is the initial velocity (what we want to find),
- g is the acceleration due to gravity (-9.8 m/s^2),
- t is the time (2 seconds in this case).

Using these two equations, we can solve for both the initial velocity (vi) and the total time of flight (t).

To find vi, we substitute the values into the displacement equation:

42 = vi(2) + 0.5(-9.8)(2)^2

Simplifying, we get:

42 = 2vi - 19.6

2vi = 42 + 19.6
2vi = 61.6
vi = 30.8 m/s

So, the arrow was shot with an initial velocity of 30.8 m/s.

To find the total time of flight, we can double the time it took to reach the highest point since the ascent and descent times are equal. Thus, the total time of flight is:

Total time = 2t
Total time = 2(2)
Total time = 4 seconds

Therefore, the arrow is in the air for a total of 4 seconds.