Question

the coordinates of a bird flying in the xy plane are given by ( x(t) = alpha t ) and ( y(t) = 3.0 , \text{m} - \beta t^2 ), where ( alpha = 2.4 , \text{m/s} ) and ( \beta = 1.2 , \text{m/s}^2 ). for help with math skills, you may want to review: differentiation of polynomial functions, vector magnitudes, determining the angle of a vector. for general problem - solving tips and strategies for this topic, you may want to view a video tutor solution of calculating average and instantaneous accelerations. part e not fully visible part f calculate the direction of the bird’s acceleration at ( t = 2.0 , \text{s} ). express your answer in degrees. view available hint(s) ( \theta = ) text box with buttons

Explanation:

Step1: Find acceleration components

The position functions are \( x(t) = \alpha t \) and \( y(t) = 3.0\ \text{m} - \beta t^2 \). To find acceleration, we take the second derivative of each position function.
For \( x(t) \):
First derivative (velocity): \( v_x(t) = \frac{dx}{dt} = \alpha \)
Second derivative (acceleration): \( a_x(t) = \frac{dv_x}{dt} = 0 \) (since \( \alpha \) is constant)

For \( y(t) \):
First derivative (velocity): \( v_y(t) = \frac{dy}{dt} = -2\beta t \)
Second derivative (acceleration): \( a_y(t) = \frac{dv_y}{dt} = -2\beta \)

Step2: Substitute values at \( t = 2.0\ \text{s} \)

Given \( \beta = 1.2\ \text{m/s}^2 \), substitute into \( a_y \):
\( a_y = -2(1.2\ \text{m/s}^2) = -2.4\ \text{m/s}^2 \)
\( a_x = 0\ \text{m/s}^2 \) (from Step 1)

Step3: Determine the direction of acceleration

The acceleration vector is \( \vec{a} = (a_x, a_y) = (0, -2.4\ \text{m/s}^2) \). This vector points along the negative \( y \)-axis. The angle \( \theta \) (measured from the positive \( x \)-axis, counterclockwise) for a vector along the negative \( y \)-axis is \( 270^\circ \) (or equivalently, \( -90^\circ \), but typically we use the range \( 0^\circ \) to \( 360^\circ \)).

Answer:

\( \boxed{270^\circ} \) (or \( \boxed{-90^\circ} \), but \( 270^\circ \) is more standard for this context)