Parametric Circular Orbits Quiz: Parametric Equations of Circular Orbits

The satellite is at (9, 0) when 3t = 2πk. Smallest positive solution: t = 2π/3. Option B gives π/3 where the satellite is at (-9, 0). Option C gives t = π also giving (-9, 0). Option D gives the second return.

The answer is True. At t = 0 the position is (R cos(p) + h, R sin(p) + k). Increasing p by π/2 changes the initial angle to p + π/2, rotating the starting point 90 degrees counterclockwise. Radius, angular speed, center, and shape are all unchanged.

Speed v = R times the absolute value of w is constant since R and w do not change with time, confirming A. Centripetal acceleration always points toward the center at (h, k), confirming B. Eliminating the parameter t gives the circle equation confirming C. Option D is false because the correct period formula is T = 2π divided by the absolute value of w, not R divided by 2π.

Speed v = R times w = 6 times 1.5 = 9 units per second. Arc length s = v times delta t = 9 times 2 = 18 units. Option B gives the speed alone without multiplying by the time interval. Option C gives R times delta t = 6 times 2 = 12, using radius instead of speed. Option D gives just the radius R = 6 with no calculation applied.

Translations -1 and +4 do not affect w. T = 2π divided by 2 = π seconds. Option B requires w = 1. Option C requires w = 0.5. Option D requires w = 4.

R = 10, w = 0.2, p = -π/4, h = -3, k = 5 matches option A. Option B has wrong center and wrong phase. Option C omits the phase. Option D uses R = 5 instead of 10.

The answer is False. Eliminating t gives (x/2) squared plus (y/3) squared = 1, which is an ellipse with semi-axes 2 and 3. A circle requires equal coefficients for cosine and sine.

w is the coefficient of t inside the trig functions. The argument is πt, so w = π rad/s. Option B gives R = 12. Option C gives 2π, correct only if the argument were 2πt. Option D gives 1/π, the frequency not angular speed.

Speed v = R times the absolute value of w depends only on R and w. A larger R at the same angular speed produces higher speed, confirming A. A higher w means faster travel around the orbit, confirming B. Phase p only shifts the starting point without affecting speed. Translations h and k relocate the center without changing the speed.

At t = 0 the angle is π/2. cos(π/2) = 0 and sin(π/2) = 1, so (x,y) = (0, 5). Option A corresponds to angle 0. Option B corresponds to angle -π/2. Option D corresponds to angle π.

In x = R cos(wt) and y = R sin(wt), the radius is R. Here R = 10 and w = 2. Option B gives w = 2, the angular speed not the radius. Option C gives R divided by 2 = 5. Option D gives R times w = 20, the speed magnitude not the radius.

The answer is True. Acceleration magnitude a = R times w squared = 7 times 1 squared = 7 units per second squared. This centripetal acceleration always points toward the center. If w were 2, acceleration would be 7 times 4 = 28.

cos(π/2) = 0 and sin(π/2) = 1, so (x,y) = (0, 4). Option A corresponds to t = π. Option B corresponds to t = 0. Option D corresponds to t = 3π/2.

The orbit radius equals R. Increasing R directly enlarges the orbit, confirming A. Multiplying both cosine and sine by 2 doubles R, confirming D. Decreasing w changes period but not radius. Changing p only shifts the starting point. Translating (h, k) moves the center without changing the radius.

T = 2π divided by w = 2π divided by 0.5 = 4π seconds. Option B corresponds to w = 1. Option C corresponds to w = 2. Option D gives 2π times R, confusing radius with angular speed.

The form x = R cos(t) + h and y = R sin(t) + k places the center at (h, k). Here h = 4 and k = -2, giving center (4, -2). Option A repeats the radius as a pair. Option C negates both values. Option D ignores the translations.

The answer is True. As t increases, the angle wt + p increases when w is greater than 0. Increasing angle corresponds to counterclockwise motion. At t = 0 the angle is p, so the satellite starts at (R cos(p), R sin(p)) and moves counterclockwise.

Speed v = R times w = 8 times 0.25 = 2 units per second. Option A gives only w = 0.25 without multiplying by R. Option B gives 1 which has no direct relationship to the given parameters. Option C gives R divided by w = 8 divided by 0.25 = 32, incorrectly dividing instead of multiplying. Only option D correctly applies v = R times w with R = 8 and w = 0.25.

At t = 0 the angle is π/3. cos(π/3) = 1/2 and sin(π/3) = √3/2, giving (x,y) = (5/2, 5√3/2). Option A swaps x and y. Option C corresponds to angle π/2. Option D corresponds to angle π.

w = 2 rad/s. T = 2π divided by w = 2π divided by 2 = π seconds. Option B requires w = 1. Option C corresponds to w = 4. Option D confuses radius with angular speed.