If the sprinter from the previous problem accelerates at that rate for 20 m, and then maintains

If the sprinter from the previous problem accelerates at...

Key Concepts

Segmented Motion Analysis

Segmented motion analysis involves breaking down a complex motion into separate stages or segments (such as an accelerating phase followed by constant speed motion). By analyzing each segment independently using appropriate kinematic relationships and then combining the results, one can accurately determine overall quantities like total time or displacement for the complete motion.

Kinematic Equations

Kinematic equations are a set of mathematical formulas that describe the relationships between displacement, velocity, acceleration, and time for objects in motion. These equations are crucial for solving problems involving both accelerating and constant velocity phases, as they provide the tools needed to compute distances, speeds, and time intervals.

Uniform Acceleration

Uniform acceleration refers to motion where the velocity of an object changes by equal amounts in equal time intervals. This concept is essential when analyzing scenarios where an object, such as a sprinter, increases speed steadily over a given distance. Using the formulas associated with uniformly accelerated motion allows for the determination of the final velocity and time taken during this phase of the movement.

Constant Velocity Motion

Constant velocity motion implies that an object moves at a steady speed without any acceleration. In this phase, the time taken to cover a certain distance is simply the distance divided by the velocity. This concept is key when the object, after a period of acceleration, attains and maintains its top speed for the rest of its journey.

Transcript

00:02 We are asked to calculate for the sprinter's time for the entire 100 meter days.

00:10 Upon reading the problem, you will see that the sprinter's motion was not constant all throughout the race.

00:17 Hence, we can divide it into two segments.

00:21 For the first segment, the sprinter has an acceleration of 4 .20 meters per second squared, that's with reference to problem number one.

00:32 For 20 meters.

00:36 And for the second segment, the sprinter maintained his current velocity for 80 meters.

00:43 So how did i know that it's for 80 meters since the race is 100 meters in total, and the first part is 20 meters.

00:52 Therefore, the second part is 80 meters.

00:56 So for us to see the situation, clearly so let's list down the given for each segment of the sprinter's motion so for segment one again so for segment one the sprinter has an acceleration of 4 .20 meters per second squared for 20 meters so since segment one is the start of the race so we can assume that everybody starts from res in a race so let's label let's label it v1 1 so b1 stands for the initial velocity and then the other one stands for segment 1 so for us to distinguish the given from segment 1 and segment 2 so we're right here zero and then he has an acceleration of 4 .20 meters per second square so that's a sub 1 again which stands for segment 1, so 4 .20 meters per second squared.

02:11 And then lastly, the distance or the displacement is 20 meters.

02:18 So let's write right here, delta x1 equals 20 meters.

02:28 And we're looking for the time.

02:33 So time 1.

02:34 So since we're looking, we were asked to calculate for the entire time, for the sprinter's time to finish the entire 100 meters.

02:46 So for segment two, so what do we know about segment two? so again, in the problem, it's said that the sprinter maintained his current velocity for 80 meters.

03:02 So we don't know anything else.

03:07 But the distance or displacement.

03:12 So let's write here, delta x sub 2, 80 meters.

03:18 But we can get another given from the statement.

03:25 So it said that the sprinter maintained his current velocity.

03:30 So maintained his current velocity, so meaning the acceleration for the second segment is zero.

03:37 So there's the the sprinter did not accelerate for the second segment and of course we're also looking for the time for this segment so we need to know the time for segment one and segment two for us to know the total time spent by the sprinter during the entire 100 meter race so since for the first segment as you can see we have four given sorry three given and one unknown we will already be able to solve for the time since we already have three given values so we just have to choose the appropriate equation from our uniformly accelerated motion equations so we have four so how do we choose which one to use so the equation that we should choose should have all the three given values so which are the initial velocity v1 the acceleration delta x and the time so what equation is that so the appropriate equation to use would be this one delta x equals v1 times time plus one half a t squared so why did we choose this because this is the only equation that doesn't have v sub 2 or the final below because we don't have that given for segment 1.

05:12 So using this we can solve for e sub 1.

05:16 So let's just write the subscripts for us to distinguish these quantities.

05:28 Alright, so we can further, so we need to solve the time using this equation.

05:37 So we can further simplify it because as you can see, see our v sub 1 is 0 so we can simplify this equation so since v1 is 0 so this will be 0 so we'll be left with a much simpler equation delta x sub 1 equals 1 half a 1 t1 square so let's arrive t from this equation so of course we have to move everything or we have to divide both sides by one half a so that t squared will be left on one side so this will be i will just transfer t on the left side so t one squared equals so dividing both sides by one half a one so that will give us two delta x of one over a one right and then since we only need t and we have here t squared, so we just have to get the square root of both sides of the equation.

06:52 So this will give us the equation for t1, which is the square root of 2 delta x sub 1 over a1.

07:03 So we can now solve for t1...

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