How long will it take Michael to catch Robert in a long distance race?

In summary, the problem is to find the distance between two people who are running at different speeds. Michael is running at 3.8 m/s and is 75 m behind Robert, who is running at a constant velocity of 4.2 m/s. If Michael accelerates at 0.15 m/s^2, how long will it take him to catch Robert. Michael's distance is complicated because of the initial velocity and acceleration. He uses the standard distance formula for accelerated motion: d = Vi*t + .5*a*t². The equations for Robert and Michael are found by solving the quadratic equation for t.
  • #1
bcaie16
5
0

Homework Statement


In a long distance race, Michael is running at 3.8 m/s and is 75 m behind Robert, who is running at a constant velocity of 4.2 m/s. If Michael accelerates at 0.15 m/s^2, how long will it take him to catch Robert

Homework Equations


D=vi(delta)t + 1/2a(delta)t^2
etc

The Attempt at a Solution


I know how to do this type of problem when the accelerating runner starts from a rest, but I have no idea where to go from here. I've tried framing the problem a bunch of different ways but nothing.
 
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  • #2
welcome to pf!

hi bcaie16! welcome to pf! :smile:

(have a delta: ∆ and try using the X2 and X2 icons just above the Reply box :wink:)

you need to write two separate equations, one for each runner …

what do you get? :smile:
 
  • #3
Welcome to PF, bcaie16! Nice to see some high school work here.
The thing that determines the answer is that the two people meet - their distances become equal. So you should begin with
Michael's distance = Robert's distance

Robert's distance is easy - just use d = vt and add 75 because he is starting out that far ahead.

Michael's distance is complicated because of the initial velocity and acceleration. Use the standard distance formula for accelerated motion: d = Vi*t + .5*a*t².

It looks like you will have a quadratic equation to solve. If you are just starting grade 11 math you may need to look up how to solve a quadratic on a calculator or using the general formula:
http://en.wikipedia.org/wiki/Quadratic_equation

You can, of course, solve it by trial and error. A spreadsheet works really well for that.
 
  • #4
I figured I would have to use the quadratic formula. I wasn't sure how to frame it kinematics wise. What are the a, b, and c values respectively in terms of velocity, acceleration, etc.
 
  • #5
Ok i figured that the respective equations would be ∆t= 4.2 ∆D +75, and
75= -∆t +/- square root ∆t -2.28/2
 
  • #6
bcaie16 said:
Ok i figured that the respective equations would be ∆t= 4.2 ∆D +75, and
75= -∆t +/- square root ∆t -2.28/2

nooo :redface:

(btw, why are you using all these ∆s? it's not normal, and it really doesn't help)

∆t= 4.2 ∆D +75 is completely wrong, isn't it?

and i can't work out what your second equation is :confused:

just write two general equations, one for each runner (ignoring the other) …

when you have them, then you can decide how to use them​
 
  • #7
Ok being completely new to physics, I am kind of lost. I am not really sure to how to get a general equation. Should they be in relation to time or distance
 
  • #8
both …

each should be x = a function of t

(ie it tells you where it is at each time t :wink:)
 
  • #9
Michael's distance = Robert's distance
vt + .5*a*t² = vt + 75
3.8*t + .5*(.15)*t² = 4.25 + 75
collect like terms, all on one side, so you have an equation of the form
at² + bt + c = 0
Then use one of the methods of solving the quadratic for t.
 
  • #10
Oops, forgot a t in the previous post. Too late to edit. Should be like this:
Michael's distance = Robert's distance
vt + .5*a*t² = vt + 75
3.8*t + .5*(.15)*t² = 4.2t + 75
collect like terms, all on one side, so you have an equation of the form
at² + bt + c = 0
Then use one of the methods of solving the quadratic for t.
 

1. What is Kinematics?

Kinematics is a branch of physics that studies the motion of objects without considering the factors that cause the motion, such as forces and energy. It focuses on describing the position, velocity, and acceleration of objects.

2. What is the difference between speed and velocity?

Speed is a scalar quantity that measures how fast an object is moving, while velocity is a vector quantity that measures the speed and direction of an object's motion. This means that velocity takes into account the direction of motion, while speed does not.

3. What is the difference between distance and displacement?

Distance is a scalar quantity that measures the total length of the path traveled by an object, while displacement is a vector quantity that measures the shortest distance between the starting and ending points of an object's motion. This means that displacement takes into account the direction of motion, while distance does not.

4. How is acceleration related to velocity?

Acceleration is the rate of change of an object's velocity over time. This means that if an object's velocity is changing, it is experiencing acceleration. If an object is moving with a constant velocity, its acceleration is zero.

5. What are the equations of motion used in kinematics?

The equations of motion used in kinematics are:

  • Position (x) = Initial position (x0) + Initial velocity (v0) * Time (t) + 1/2 * Acceleration (a) * Time (t)2
  • Velocity (v) = Initial velocity (v0) + Acceleration (a) * Time (t)
  • Acceleration (a) = Change in velocity (v - v0) / Time (t)
  • Final velocity (v) = Square root of (Initial velocity (v0)2 + 2 * Acceleration (a) * Displacement (x - x0))
  • Displacement (x) = Initial position (x0) + 1/2 * (Initial velocity (v0) + Final velocity (v)) * Time (t)

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