During a track event two runners, Mary, and Alice, round the last turn and head into the final stretch with Mary a distance d = 4.0 m in front of Alice. They are both running with the same speed of v0= 7.0 m/s. When the finish line is a distance L= 45.0 m away from Alice, Alice accelerates at aA = 0.5 m/s2 until she catches up to Mary. Alice then continues at a constant speed until she reaches the finish line.
(a) How long (in s) did it take Alice to catch up with Mary?
(b) How far (in m) did Alice still have to run when she just caught up to Mary?
(c) How long (in s) did Alice take to reach the finish line after she just caught up to Mary?
Mary starts to accelerate when Alice just catches up to her, and accelerates all the way to the finish line and crosses the line exactly when Alice does. Assume Mary's acceleration is constant.
(d) What is Mary's acceleration (in m/
(e) What is Mary's velocity at the finish line (in m/s)?
a.4
b.13
c.1.44
d.2.82
e.11.06
To find out when Alice catches up to Mary and how long it takes, we need to analyze their motion and determine the distance each runner travels.
Let's start by determining the time it takes for Alice to catch up to Mary.
We know that Alice is initially d = 4.0 m behind Mary and both runners have the same speed, v0 = 7.0 m/s. Since Alice starts accelerating after a certain time, let's call the time it takes for Alice to catch up to Mary as t1.
During this time t1, Mary continues to run at a constant speed of v0 = 7.0 m/s. The distance Mary travels during time t1 can be calculated using the formula:
Distance = Speed x Time
Distance Mary travels = v0 × t1
Since both runners have the same speed, Alice also travels the same distance as Mary during time t1. However, Alice starts with a distance of d = 4.0 m less than Mary. So, the distance Alice travels during time t1 can be calculated as:
Distance Alice travels = Distance Mary travels - 4.0 m
Next, Alice starts accelerating at a constant acceleration aA = 0.5 m/s^2 until she catches up to Mary. To determine the time it takes for Alice to catch up to Mary after accelerating, we can use the kinematic equation:
Distance = Initial velocity × Time + 0.5 × Acceleration × Time^2
In this case, the initial velocity is v0 = 7.0 m/s, and the acceleration is aA = 0.5 m/s^2. The distance Alice needs to cover to catch up to Mary is the initial distance d = 4.0 m.
So, using the kinematic equation, we can write:
d = v0 × t2 + 0.5 × aA × t2^2
Simplifying the equation:
4.0 m = 7.0 m/s × t2 + 0.5 m/s^2 × t2^2
This is a quadratic equation, which we can solve to find the time t2.
Once Alice catches up to Mary, she continues at a constant speed until reaching the finish line, which is a distance L = 45.0 m away from Alice.
To find the time it takes for Alice to reach the finish line from the point she catches up to Mary (t3), we can use the formula:
Distance = Speed × Time
45.0 m = Speed × t3
Since Alice continues at a constant speed, the speed she needs to use is v0 = 7.0 m/s.
Now, we have the time t1, t2, and t3, which are the times it takes for Alice to catch up to Mary, for Alice to accelerate, and for Alice to reach the finish line, respectively.
To calculate the total time it takes for Alice to finish the race, we need to add up these three times:
Total Time = t1 + t2 + t3