Question

001 (part 1 of 2) 10.0 points a runner is jogging in a straight line at a steady v_r = 2.3 km/hr. when the runner is l = 2.1 km from the finish line, a bird begins flying straight from the runner to the finish line at v_b = 6.9 km/hr (3 times as fast as the runner). when the bird reaches the finish line, it turns around and flies directly back to the runner. what cumulative distance does the bird travel? even though the bird is a dodo, assume that it occupies only one point in space (a “zero” length bird), travels in a straight line, and that it can turn without loss of speed. answer in units of km.

Explanation:

Step1: Calculate time for bird to reach finish - line

The time $t_1$ it takes for the bird to reach the finish - line is given by the formula $t_1=\frac{L}{v_b}$, where $L = 2.1$ km and $v_b=6.9$ km/hr. So $t_1=\frac{2.1}{6.9}$ hr.

Step2: Calculate distance runner moves in $t_1$

The distance $d_r$ the runner moves in time $t_1$ is $d_r = v_r\times t_1$, with $v_r = 2.3$ km/hr. Substituting $t_1=\frac{2.1}{6.9}$ hr, we get $d_r=2.3\times\frac{2.1}{6.9}=0.7$ km.

Step3: Calculate distance between bird and runner when bird reaches finish - line

The distance $d$ between the bird and the runner when the bird reaches the finish - line is $d = L - d_r=2.1 - 0.7 = 1.4$ km.

Step4: Calculate time for bird - runner meeting

The relative speed $v_{rel}$ of the bird and the runner when they are moving towards each other is $v_{rel}=v_r + v_b=2.3+6.9 = 9.2$ km/hr. The time $t_2$ it takes for them to meet is $t_2=\frac{d}{v_{rel}}=\frac{1.4}{9.2}$ hr.

Step5: Calculate distance bird moves in $t_2$

The distance $d_{b2}$ the bird moves in time $t_2$ is $d_{b2}=v_b\times t_2=6.9\times\frac{1.4}{9.2}=1.05$ km.

Step6: Calculate total distance bird travels

The total distance $D$ the bird travels is the distance to the finish - line plus the distance back. The distance to the finish - line is $L = 2.1$ km, and the distance back is $d_{b2}=1.05$ km. So $D=2.1 + 1.05=3.15$ km.

Answer:

$3.15$ km