When Mary is 22 m from the finish line, she has a speed of 4.0 m/s and is 5.0 m behind Sally, who has a speed of 5.0 m/s. Sally thinks she has an easy win and so, during the remaining portion of the race, decelerates at a constant rate of 0.51 m/s^2 to the finish line.
What constant acceleration does Mary now need during the remaining portion of the race, if she wishes to cross the finish line side-by-side with Sally?
So they cross at the same time.
Sally:
d=Vi*t+1/2 at^2 where a=-.51m/s^2 d=22-7
solve for time t.
Mary:
d=22=Vi*t-1/2 a t^2 solve for a, using t above.
d=22-5
not d=22-7
To determine the constant acceleration that Mary needs during the remaining portion of the race, we need to first find out the time it will take for Sally to reach the finish line after Mary.
We know that Mary is initially 5.0 m behind Sally when Mary is 22 m from the finish line. Therefore, Sally has to cover a distance of 22 + 5 = 27 m to reach the finish line.
Next, we can calculate the time it will take for Sally to reach the finish line using her initial speed of 5.0 m/s and her constant deceleration of 0.51 m/s^2. We can use the following equation to calculate the time:
v_f = v_i + at
Where:
v_f = final velocity (0 m/s since Sally comes to rest at the finish line)
v_i = initial velocity (5.0 m/s)
a = acceleration (-0.51 m/s^2)
t = time
0 = 5.0 - (0.51 × t)
Solving for t:
0.51 × t = 5.0
t ≈ 9.80 s
So, it takes Sally approximately 9.80 seconds to reach the finish line after Mary.
Now, we can find out the constant acceleration that Mary needs during the remaining portion of the race so that she crosses the finish line side-by-side with Sally.
The equation to calculate the final velocity can be used:
v_f = v_i + at
Where:
v_f = final velocity (unknown)
v_i = initial velocity (4.0 m/s for Mary)
a = acceleration (unknown)
t = time (9.80 s)
To cross the finish line side-by-side with Sally, Mary's final velocity needs to be 0 m/s, just like Sally. Thus, the equation becomes:
0 = 4.0 + (a × 9.80)
Solving for a:
a × 9.80 = -4.0
a ≈ -0.408 m/s^2
Therefore, Mary needs a constant acceleration of approximately -0.408 m/s^2 during the remaining portion of the race in order to cross the finish line side-by-side with Sally.