What equations of motion under constant acceleration do you know? There are a standard five "SUVAT" equations.heythere1010 said:Homework Statement
Justin is racing Mark. Justin accelerates from rest at the starting point at 1.2 m/s^2 [E]. Mark is still adjusting his equipment and 2.0 s later begins to accelerate at 1.5 m/s^2 [E]. Where and when does Mark catch up to Justin?
Homework Equations
The Attempt at a Solution
I think I need to create two formulas, set them equal to each other, and use the quadratic formula. I'm not sure how to create the formulas though.
v=u+at, d=ut+1/2at^2, d=vt-1/2at^2haruspex said:What equations of motion under constant acceleration do you know? There are a standard five "SUVAT" equations.
Ok. You can see that there are five variables that recur, two of speed, one acceleration, one distance, and one time. Which of those five is of no interest here? Find the equation that does not involve that variable.heythere1010 said:v=u+at, d=ut+1/2at^2, d=vt-1/2at^2
So d=ut+1/2at^2.haruspex said:Ok. You can see that there are five variables that recur, two of speed, one acceleration, one distance, and one time. Which of those five is of no interest here? Find the equation that does not involve that variable.
Right. So write out that equation for each runner, and the relationships between the times and distances in the two equations.heythere1010 said:So d=ut+1/2at^2.
How would I set them up if there are two unknown variables t and d?haruspex said:Right. So write out that equation for each runner, and the relationships between the times and distances in the two equations.
There'll be two unknowns in each equation, but in each case there's a known relationship between the unknown in one equation and the corresponding unknown in the other equation. So you have enough equations to solve for all unknowns.heythere1010 said:How would I set them up if there are two unknown variables t and d?
Which variables are related?haruspex said:There'll be two unknowns in each equation, but in each case there's a known relationship between the unknown in one equation and the corresponding unknown in the other equation. So you have enough equations to solve for all unknowns.
The accelerations are known to be different, and the initial velocities are both known. What does that leave?heythere1010 said:Which variables are related?
Distance. But I don't think the equation would be d=0.6t^2 and d=0.75t^2 right?haruspex said:The accelerations are known to be different, and the initial velocities are both known. What does that leave?
That's right, because the the times are different. But they are related - how?heythere1010 said:Distance. But I don't think the equation would be d=0.6t^2 and d=0.75t^2 right?
The distance is the same.haruspex said:That's right, because the the times are different. But they are related - how?
Sure, but how are the two times related? You are told.heythere1010 said:The distance is the same.
Yes, one is two seconds longer, t+2.haruspex said:Sure, but how are the two times related? You are told.
Right. So what equations do you have now?heythere1010 said:Yes, one is two seconds longer, t+2.
I don't know where t+2 would go.haruspex said:Right. So what equations do you have now?
Simplest way to think about it is this: let t1 be the first runner's acceleration time and t2 be the second runner's acceleration time. That allows you to write the two SUVAT equations. Now you just have to relate t1 and t2 by the fact that they differ by 2 - the tricky part is to make sure you get it the right way round. Think: which runner runs for longer?heythere1010 said:I don't know where t+2 would go.
I'm still not sure how I add the 2 seconds into the equation.haruspex said:Simplest way to think about it is this: let t1 be the first runner's acceleration time and t2 be the second runner's acceleration time. That allows you to write the two SUVAT equations. Now you just have to relate t1 and t2 by the fact that they differ by 2 - the tricky part is to make sure you get it the right way round. Think: which runner runs for longer?
If Justin (first runner) runs for t1 seconds, how long does Mark run for?heythere1010 said:I'm still not sure how I add the 2 seconds into the equation.
t-2.haruspex said:If Justin (first runner) runs for t1 seconds, how long does Mark run for?
Right, so plug t-2 in place of t in Mark's equation.heythere1010 said:t-2.
Yes, I did this, but when I make it equal to the other equation, both my answers are positive.haruspex said:Right, so plug t-2 in place of t in Mark's equation.
Please post your working.heythere1010 said:Yes, I did this, but when I make it equal to the other equation, both my answers are positive.
0.75(t-2)^2=0.6t^2haruspex said:Please post your working.
It is quite possible they are both positive, but is one less than 2 seconds?
OK, but with Mark running for only t-2 seconds, what do you think t=1.1 means?heythere1010 said:0.75(t-2)^2=0.6t^2
Solving using the quadratic formula gave me 18.9 and 1.1.
Right haha. Really late where I am. Thanks for the help!haruspex said:OK, but with Mark running for only t-2 seconds, what do you think t=1.1 means?
To calculate the acceleration of a runner, you need to know their initial velocity, final velocity, and the time it takes for them to travel between those two points. The formula for acceleration is (final velocity - initial velocity)/time.
Acceleration is the rate of change of velocity, while velocity is the speed and direction of an object's motion. In simpler terms, acceleration is how quickly an object's velocity is changing, while velocity is how fast and in what direction an object is moving.
The acceleration of a runner can greatly impact their race. A runner with a higher acceleration will be able to increase their speed faster and ultimately finish the race quicker. However, a runner with a lower acceleration may have a more consistent pace and be able to maintain their speed for a longer period of time.
Yes, it is possible for two runners with different acceleration rates to have the same final velocity. This can happen if the runner with the lower acceleration starts with a higher initial velocity or if the runner with the higher acceleration starts with a lower initial velocity.
If two runners have different acceleration rates, the distance between them will change over time. The runner with the higher acceleration will catch up to and eventually pass the runner with the lower acceleration. The distance between them will decrease as the faster runner gains speed and the slower runner falls behind.