Some rental cars have a GPS unit installed, which allows the rental car company to check where you are at all times and thus also know your speed at any time. One of these rental cars is driven by an employee in the company’s lot and, during the time interval from 0 to 10 s, is found to have a position vector as a function of time of
(t) = ( (12.9 m) - t(16.9 m/s) + t2(2.13 m/s2) , (58.1 m) + t2(2.29 m/s2) - + t3(0.113 m/s3) )
a) What is the distance of this car from the origin of the coordinate system at t = 5.50 s?
b) What is the velocity vector at t = 5.50 s? [Enter the x-component into the first box, the y-component into the second.]
c) What is the speed at t = 5.50 s?
To find the distance of the rental car from the origin at t = 5.50 s, we need to calculate the magnitude of the position vector at that time.
a) The magnitude of a position vector can be found using the Pythagorean theorem: magnitude = sqrt(x^2 + y^2), where x and y are the components of the position vector.
1. Plug in t = 5.50 s into the position vector equation:
r(t) = (12.9 m - 5.5(16.9 m/s) + (5.5^2)(2.13 m/s^2), 58.1 m + (5.5^2)(2.29 m/s^2) - + (5.5^3)(0.113 m/s^3))
2. Calculate the x and y components of the position vector by evaluating the expressions in the brackets:
x = 12.9 m - 5.5(16.9 m/s) + (5.5^2)(2.13 m/s^2)
y = 58.1 m + (5.5^2)(2.29 m/s^2) - + (5.5^3)(0.113 m/s^3)
3. Square each component and add them together:
magnitude = sqrt(x^2 + y^2)
b) To find the velocity vector at t = 5.50 s, we need to differentiate the position vector with respect to time:
v(t) = (dx/dt, dy/dt)
1. Differentiate each component of the position vector:
dx/dt = -16.9 m/s + 2(5.5)(2.13 m/s^2)
dy/dt = 2(5.5)(2.29 m/s^2) - 3(5.5^2)(0.113 m/s^3)
2. Plug in t = 5.50 s and calculate the values for dx/dt and dy/dt.
c) The speed at t = 5.50 s is the magnitude of the velocity vector at that time:
speed = sqrt((dx/dt)^2 + (dy/dt)^2)
1. Calculate the values of dx/dt and dy/dt as in step 2 of part b.
2. Use the values to calculate the speed using the formula above.
By following these steps, you should be able to find the answers to parts a, b, and c of the problem.