Question

A remote-controlled car is moving in a vacant parking lot. The velocity of the car as a function of time is given by 𝑣 = [5.00𝑚/𝑠 − (0.0180𝑚/ 𝑠 3 )𝑡 2 ]𝑖̂+ [2.00 𝑚⁄𝑠 + (0.550 𝑚 𝑠 2 ⁄ )𝑡]𝑗. (a) What are ̂ 𝑎𝑥(𝑡) and 𝑎𝑦(𝑡), the 𝑥- and 𝑦-components of the car’s velocity as functions of time? (b) What are the magnitude and direction of the car’s velocity at 𝑡 = 8.00 𝑠? (b) What are the magnitude and direction of the car’s acceleration at 𝑡 = 8.00 𝑠?

Answer 1

(a) The x- and y-component of car's velocity:

$$v_y=[2.00+ (0.550 )𝑡] \text{ m/s},\\ v_x=[5.00 − (0.0180)𝑡^2 ]\text{ m/s}.$$

Find the acceleration:

$$a(t)=v'(t)=[-0.036t]\hat i+[0.55]\hat j.\\ a_x(t)=-0.036t,\\ a_y(t)=\text{const} =0.55\text{ m/s}^2.$$

(b) The magnitude:

$$v=\sqrt{(2+0.55·8)^2+(5-0.018·8^2)^2}=7.47\text{m/s},\\\space\\ \theta=\arctan\frac{2+0.55·8}{5-0.018·8^2} =58.0°$$

from +i toward +j.

$$a=\sqrt{(-0.036·8)^2+(0.55)^2}=0.620\text{ m/s}^2.\\\space\\ \phi=\arctan\frac{0.55}{-0.036·8}=-62.4°$$

from +i toward +j (or 62.4° from +i to +j).