Throwing a Ball

Alignments to Content Standards: A-CED.A.2

A ball thrown vertically upward at a speed of $v$ ft/sec rises a distance $d$ feet in $t$ seconds, given by $$d = 6 + vt - 16t^2$$

Write an equation whose solution is

1. The time it takes a ball thrown at a speed of 88 ft/sec to rise 20 feet.

2. The speed with which the ball must be thrown to rise 20 feet in 2 seconds.

IM Commentary

Although this task is quite straightforward, it has a couple of aspects designed to encourage students to attend to the structure of the equation and the meaning of the variables in it:

• By focusing on different variables in parts (a) and (b), it fosters flexibility in seeing the same equation in two different ways: first as an equation in t with constants v and d, then as an equation in v with constants t and d.
• It does not give the values of the constants by means of an explicit equation such as $v=88$ or $d=20$. Thus it requires students to attend to the meaning of the variables in the preamble and extract the values from the descriptions in parts (a) and (b).

(Task from Algebra: Form and Function, McCallum et al., Wiley 2010)

Solution

We have an equation in three unknowns, and each part specifies two of them giving an equation in one unknown whose solution gives the desired quantity.

1. We want $d = 20$, and we are given $v = 88$, so the equation is $20 = 6 + 88t - 16 t^2$.

2. We want $d = 20$, and we are given $t = 2$, so the equation is $20 = 6 + 2v - 16\cdot 2^2$. Simplifying the right-hand side, we get $20 = -58 + 2v$.