1.
Answer the following questions. Use Equation Editor to write mathematical expressions and equations. First, save this file to your hard drive by selecting Save As from the File menu. Click the white space below each question to maintain proper formatting.
1. The gravitational force between two large space ships in space may be expressed as
a. Solve the equation for r
b. Suppose that two large ships experience a force of 0.00667 Newtons when they are 10 meters apart, find the value of G that make the relationship valid.
c. Use the value of G you found in the previous question to determine how strong the force would be when the ships are
i. 100 meters apart
ii. 40 meters apart
1. The time (sec) it takes for an object to fall from a height of h(ft) is expressed by the following equation:
a. Solve this equation for t.
b. A penny dropped from the top of the Grand Canyon takes 19.4 seconds to hit the ground at the bottom of the canyon; how deep is the canyon?
To answer these questions, we need to manipulate the given equations and solve for the desired variables.
a. To solve the equation F = G * (m1 * m2) / r^2 for r, we can rearrange the equation as follows:
F * r^2 = G * (m1 * m2)
r^2 = (G * (m1 * m2)) / F
r = sqrt((G * (m1 * m2)) / F)
b. Given that the force F is 0.00667 Newtons when the two ships are 10 meters apart, we can substitute these values into the equation and solve for G:
0.00667 * 10^2 = G * (m1 * m2)
0.00667 * 100 = G * (m1 * m2)
0.667 = G * (m1 * m2)
From this equation, it is not possible to find the value of G without knowing the masses of the two ships (m1 and m2).
c. Once we have the value of G from the previous question, we can use it to calculate the force when the ships are at different distances:
i. When the ships are 100 meters apart:
F = G * (m1 * m2) / r^2
F = G * (m1 * m2) / 100^2
ii. When the ships are 40 meters apart:
F = G * (m1 * m2) / r^2
F = G * (m1 * m2) / 40^2
2. To solve the equation h = 16t^2, we can rearrange it as follows:
t^2 = h / 16
t = sqrt(h / 16)
a. We have already expressed t in terms of h.
b. Given that a penny takes 19.4 seconds to fall from the top of the Grand Canyon, we can substitute this value into the equation and solve for h:
19.4^2 = h / 16
376.36 = h / 16
h = 376.36 * 16
So, the depth of the Grand Canyon is approximately 6029.76 feet.