An Alien spacecraft is ascending vertically over Albuquerque with a speed of 10.0 m/s.  At a height of 100 m above the Earth, an object (at this height) is dropped from the craft. How much time does it take for the object to reach the ground?

State what you are trying to find.

time, t

Establish the coordinate system and positive direction

Set positive direction down and place the coordinate system at the location of the object that is dropped.

List the known quantities

y₀ = 0 because of the placement of the origin on the package

v₀ = -10.0 m/s, same as the spacecraft, negative because positive selected to be down

y = 100 m, height of package when it is dropped. Note that y is not distance, it is displacement - from the origin

a = 9.8 m/s²  positive because same direction as positive direction chosen above

Identify the appropriate equation to use, one in which the quantity you are trying to find is the only unknown

(note that this is similar to the ball thrown upwards problem worked earlier, except that the object continues to the ground)

y = y₀ + V₀t + 1/2at²  since the problem involves motion in the y direction and all variables are known except t

Solve the equation for the unknown variable

100 = 0 - (10.0)t + 1/2(9.8)t²  = -10.0t + 4.9t²    Note: This is a quadratic equation so we expect 2 solutions or roots

4.9t² - 10.0t - 100 = 0     Note: This rearrangement is done to get the equation in correct form to use the quadratic equation

remember that

    a is the coefficient of the squared term (4.9),

    b is the coefficient of the single power term (-10.0), and

    c is the constant term (-100)

Using the quadratic formula (the formula will be provided for all review exercises)

t = -b + (b² - 4ac)^1/2/2a = 10.0 ± (100.0 - 4(4.9(-100))^1/2/2(4.9)

t = (10 ±  45.4)/9.8 ð   t = 5.7 seconds and t = -3.61 seconds

Choose the positive time, t = 5.7 seconds