A student laying on a building 54.0 m high shoots a pellet gun horizontally. If the pellet lands 156 m form the base of the building, what was the muzzle velocity of the gun?

To find the muzzle velocity of the gun, we can use the kinematic equation for horizontal motion:

Range = Initial velocity * Time

Here, the horizontal range is given as 156 m and the initial vertical velocity is 0 m/s since the pellet starts from rest vertically. Also, the time of flight is the same as the time it takes for the pellet to fall from a height of 54.0 m.

To calculate the time, we can use the kinematic equation for vertical motion:

Vertical displacement = (Initial vertical velocity * Time) + (0.5 * acceleration * Time^2)

In this case, the initial vertical velocity is 0 m/s, the vertical displacement is -54.0 m (negative because the pellet is falling downward), and the acceleration due to gravity is -9.8 m/s^2.

Plugging in the values:

-54.0 = (0 * Time) + (0.5 * (-9.8) * Time^2)

Simplifying:

-54 = -4.9 * Time^2

Rearranging the equation:

Time^2 = -54 / -4.9

Time^2 = 11.020408

Time ≈ √11.020408

Time ≈ 3.321 s (rounded to three decimal places)

Now, we can substitute the time into the horizontal motion equation to find the initial velocity (muzzle velocity):

156 = Initial velocity * 3.321

Initial velocity = 156 / 3.321

Initial velocity ≈ 46.98 m/s (rounded to two decimal places)

Therefore, the muzzle velocity of the gun is approximately 46.98 m/s.