One athlete in a race runnng on a long, straight track with a constanct speed v1 is a distance d behind a second athlete running with a constant speed v2.
(a) Under what circumstances is the first athlete able to overtake the second athlete? (Use v_1 for v1, v_2 for v2, and d as appropriate.)
(b) Find the time t it takes the first athlete to overtake the second athlete, in terms of d, v1, and v2.
(c) At what minimum distance d2 from the leading athlete must the finish line be located so that the trailing athlete can at least tie for the first place? Express d2 in terms of d, v1, and v2 by using the result of part (b).
a.
Of course, one way for athlete 1 to overtake athlete 2 is for him to accelerate (that is, if v2 > v1).
But since the given are constant speeds, surely v2 MUST BE LESS THAN v1 in order for athlete 1 to overtake athlete 2, WITHOUT accelerating.
b.
Recall that the distance travelled by an object is just speed x time:
d = vt
Athlete 2 is ahead by a distance, d, and after some time, t, their distance travelled will be equal (that is, athlete 1 overtakes athlete 2):
d + (v2)(t) = (v1)(t)
Solving for t,
t = d/(v1 - v2)
c.
Now there is an additional condition. Athlete 2 is d2 away from the finish line, while athlete 1 is (d + d2) away from the finish line.
d2 = (v2)t
from b., we substitute the expression for t:
d2 = (v2)*(d)/ (v1 - v2)
Hope this helps~ :3
(a) The first athlete is able to overtake the second athlete when the distance covered by the first athlete becomes equal to or greater than the initial distance between them.
(b) To find the time it takes for the first athlete to overtake the second athlete, we can use the equation:
t = d / (v1 - v2)
where t is the time, d is the initial distance between them, v1 is the speed of the first athlete, and v2 is the speed of the second athlete.
(c) After finding the time it takes for the first athlete to overtake the second athlete in part (b), we can find the distance d2 from the leading athlete by multiplying the time t by the speed v2:
d2 = t * v2
Substituting the expression for t from part (b) into the equation:
d2 = (d / (v1 - v2)) * v2
Therefore, the minimum distance d2 from the leading athlete must be d2 = (d * v2) / (v1 - v2)
(a) The first athlete will be able to overtake the second athlete when their relative speed is greater than zero. The relative speed is the difference between their individual speeds, so the condition for overtaking is:
v1 - v2 > 0
(b) To find the time it takes for the first athlete to overtake the second, we need to determine the time it takes for them to cover the distance between them. Let's set up an equation for this:
Distance = Speed * Time
The first athlete has to cover the initial distance d, while the second athlete has to cover 0 distance since they are not moving relative to themselves. Therefore, the equation becomes:
d = (v1 - v2) * t
Solving for t, we find:
t = d / (v1 - v2)
(c) To find the minimum distance d2 from the leading athlete to the finish line, we need to determine the time it takes for the trailing athlete to reach the finish line after overtaking. Since the first athlete takes time t to overtake the second athlete, the second athlete has to cover the remaining distance d2 in the same time t. Therefore, the equation becomes:
d2 = v2 * t
Substituting the value of t from part (b), we get:
d2 = v2 * (d / (v1 - v2))
Thus, the minimum distance d2 from the leading athlete to the finish line is given by:
d2 = (d * v2) / (v1 - v2)