Question

The x-coordinates of two objects moving along the x-axis are given as a function of time (t). x1= (4m/s)t x2= -(161m) + (48m/s)t - (4 m/s^2)t^2 Calculate the magnitude of the distance of closest approach of the two objects. x1 and x2 never have the same value.

Answer 1

To calculate the magnitude of the distance of closest approach of the two objects, we need to find the minimum distance between the x1 and x2 functions. The minimum distance occurs when the derivative of the difference between x1 and x2 is zero. We can find this point by equating the derivatives of x1 and x2 and solving for t.

Step-by-step explanation:

To calculate the magnitude of the distance of closest approach of the two objects, we need to find the minimum distance between the x1 and x2 functions. The minimum distance occurs when the derivative of the difference between x1 and x2 is zero. We can find this point by equating the derivatives of x1 and x2 and solving for t. Once we have the value of t, we can substitute it back into either x1 or x2 to find the minimum distance.

1. Take the derivative of x1 with respect to t: x1'(t) = 4 m/s
2. Take the derivative of x2 with respect to t: x2'(t) = 48 m/s - 8 m/s^2 * t
3. Set x1'(t) = x2'(t) and solve for t: 4 m/s = 48 m/s - 8 m/s^2 * t
4. Solve the equation: t = 5 s
5. Substitute t=5s into x1 or x2 to find the minimum distance: x1(5s) = 20 m

Therefore, the magnitude of the distance of closest approach of the two objects is 20 meters.

Answer 2

The two displacement functions are
x₁ = 4t
x₂ = -161 + 48t - 4t²
where
x₁, x₂ are in meters
t is time, s

The distance between the two objects is
x = x₁ - x₂
= 4t + 161 - 48t + 4t²
x = 4t² - 44t + 161

Write this equation in the standard form for a parabola.
x = 4[t² - 11t] + 161
= 4[ (t - 5.5)² - 5.5² ] + 161
x = 4(t-5)² + 40

Ths is a parabola that faces up and has its vertex (lowest point) at (5, 40).
Therefore the closest approach of the two objects is 40 m.
The graph of x versus t confirms the result.

Answer: The distance of the closest approach is 40 m.