A physics student throws a softball straight up into the air. The ball was in the air for a total of 6.16 s before it was caught at its original position.

(a) What was the initial velocity of the ball?
(b) How high did it rise?

time to top = 3.08 s

v = Vi - gt
v is 0 at top
so
Vi = 9.8*3.08

h = Vi t - 4.9 t^2 where t = 3.08

46.578224

(a) Well, if the physics student threw the softball straight up, then the initial velocity would be an upwards velocity. However, since the ball was caught at its original position, the final velocity would be zero. So, considering the ball's motion was only affected by gravity, I'd say the initial velocity was a case of "going up must come down." In other words, it was probably negative!

To find the initial velocity of the ball and the maximum height it reached, we need to apply the equations of motion for projectile motion.

(a) Finding the initial velocity:
To determine the initial velocity, we can use the formula:

v = u + at

where:
v is the final velocity (which is 0 when the ball reaches its highest point),
u is the initial velocity,
a is the acceleration (which is the acceleration due to gravity, approximately -9.8 m/s²),
and t is the time taken (which is half of the total time the ball was in the air since it takes equal time to rise and fall).

Plugging in the known values:
v = 0
a = -9.8 m/s²
t = 6.16 s / 2 = 3.08 s

Now, we can rearrange the equation to solve for u:

u = v - at

u = 0 - (-9.8 m/s²) * 3.08 s
u = 30.184 m/s

Therefore, the initial velocity of the ball is approximately 30.184 m/s.

(b) Finding the maximum height:
To determine the maximum height the ball reached, we can use the formula:

h = u * t + (1/2) * a * t²

where:
h is the maximum height,
u is the initial velocity,
t is the time taken (which is half of the total time the ball was in the air, as mentioned earlier),
and a is the acceleration (which is the acceleration due to gravity).

Plugging in the known values:
u = 30.184 m/s
t = 3.08 s
a = -9.8 m/s²

Now, we can calculate the maximum height:

h = 30.184 m/s * 3.08 s + (1/2) * (-9.8 m/s²) * (3.08 s)²
h = 93.0576 m + (1/2) * (-9.8 m/s²) * 9.4864 s²
h = 93.0576 m + (-46.26048 m)
h ≈ 46.80 m

Therefore, the ball reached a maximum height of approximately 46.80 meters.