Ahistorical society is testing an old cannon. they
place the cannon in an open, level field and perform
a few test fires to determine the speed at which the
cannonballs leave the cannon. the cannon is placed
at a 45.0° angle from the horizontal and placed in a
bunker so that the cannonballs are fired from ground
level. they measure the flight time of one cannonball
to be 3.78 s. (2.3) m
(a) what is the speed of the cannonball as it leaves
the cannon?

26.2 m/s

Explanation:

We can find the speed of the cannonball just by analyzing its vertical motion. In fact, the initial vertical velocity is

(1)

where u is the initial speed and is the angle of projection.

We can therefore use the following suvat equation for the vertical motion of the ball:

where is the vertical velocity at time t, and is the acceleration of gravity. The time of flight is 3.78 s, so we know that the ball reaches its maximum height at half this time:

And at the maximum height, the vertical velocity is zero:

Substituting these values, we find the initial vertical velocity:

And using eq.(1) we now find the initial speed: