Solution

We assume the direction of motion is +x and assume the refrigerator starts from rest (so that the speed being discussed is the velocity $$\vec{v}$$ that results from the process). The only force along the x axis is the x component of the applied force $$\vec{F}$$.

(a) Since $$v_0 = 0$$, the combination of Eq. 2-11 and Eq. 5-2 leads simply to
$$F_x = m\left ( \frac{v}{t} \right )\Rightarrow v_i\left ( \frac{F\, cos \,\theta _i}{m} \right )t$$

for i = 1 or 2 (where we denote $$\theta _ 1 = \theta $$ and $$\theta _2 = \theta $$ for the two cases). Hence, we see that the ratio $$v_2$$ over $$v_1$$ is equal to $$cos\, \theta$$.

(b) Since $$v_0 = 0$$, the combination of Eq. 2-16 and Eq. 5-2 leads to
$$F_x = m \left ( \frac{v^2}{2\Delta x} \right )\Rightarrow v_i =\sqrt{2\left ( \frac{F \, cos \, \theta_i }{m} \right )}$$

for i = 1 or 2 (again, $$\theta _1 = \theta $$ and $$\theta _2 = \theta $$ is used for the two cases). In this scenario, we see that the ratio $$v_2$$ over $$v_1$$ is equal to $$\sqrt{cos\, \theta} $$.