Trajectory for a runner aiming to a fixed point, with initial velocity?

In summary, to calculate the time it would take a running runner to reach a point when initially running in the wrong direction, you can break down the problem into smaller steps and use the initial velocity, initial position, target position, acceleration, deceleration, and maximum velocity as data points. By calculating the time it would take for the runner to decelerate to zero velocity, accelerate to the maximum velocity, and travel the remaining distance at the maximum velocity, you can determine the total time it would take for the runner to reach the target point.
  • #1
ManuTOO
1
0
Hello,

I want to know how long time a running runner needs to reach a point, when he is initially running in the wrong direction.

I'd like to do this with minimal calculation, so an approximation at 10% would be enough.

To even complicate the problem more, the runner can decelerate faster than accelerate.

data used :
- initial velocity
- initial position
- target position
- acceleration
- deceleration
- maximal velocity

The runner will always accelerate in the direction of the target position, and hopefully he won't run fast enough to enter in orbit around it... ;)

I tried to look for a solution on Google & these forums, but after more than 20 min still found nothing (especially coz I'm not sure of the right keywords to use :-S )

Thanks in advance for any help..! :)
 
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  • #2
</code>The best way to approach this problem is to break it down into smaller steps to make the calculations easier. For example, you could calculate how long it would take the runner to decelerate to zero velocity in the wrong direction, and then calculate how long it would take to accelerate from zero velocity to the target point. You can also use the maximum velocity of the runner as a limiting factor, so that if the runner accelerates or decelerates too fast, they will not reach the target point.First, calculate the distance between the initial position and the target position. Next, calculate the time it would take the runner to decelerate from the initial velocity to zero velocity. This can be done by dividing the initial velocity by the deceleration rate. Then, calculate the time it would take the runner to accelerate from zero velocity to the maximum velocity. This can be done by dividing the maximum velocity by the acceleration rate. Finally, calculate the time it would take the runner to travel the remaining distance between the initial position and the target position at the maximum velocity. This can be done by dividing the remaining distance by the maximum velocity. By adding together the times calculated in the above steps, you can get the total time it would take for the runner to reach the target point.I hope this helps!
 
  • #3


Hello,

Thank you for your question. I can provide you with a response to your inquiry about the trajectory of a runner aiming for a fixed point with an initial velocity.

To calculate the time it takes for the runner to reach the fixed point, we can use the equation d = v0t + 1/2at^2, where d is the distance traveled, v0 is the initial velocity, a is the acceleration, and t is the time. In this case, we can assume that the runner starts at an initial position of 0, and the target position is a fixed point.

Using the given data of initial velocity, initial position, target position, acceleration, deceleration, and maximal velocity, we can plug in these values into the equation and solve for t. However, since you mentioned wanting to do this with minimal calculation and an approximation of 10%, we can use a simpler approach.

We can use the average velocity of the runner, which is calculated by taking the sum of the initial velocity and maximal velocity and dividing it by 2. This is a reasonable approximation since we know that the runner will accelerate and decelerate during the trajectory, so the average velocity will give us a good estimate.

Then, we can use the formula t = d/v, where d is the distance between the initial and target position, and v is the average velocity. This will give us an approximate time it takes for the runner to reach the fixed point.

I hope this helps and provides you with a solution to your problem. If you have any further questions or need clarification, please don't hesitate to ask. Best of luck with your calculations!
 

1. What is trajectory?

Trajectory refers to the path that an object takes as it moves through space, often in reference to its motion in relation to a fixed point.

2. How is trajectory calculated for a runner?

To calculate the trajectory of a runner aiming for a fixed point, you would need to take into account the runner's initial velocity, the angle at which they are running, and any external factors such as wind resistance or terrain.

3. What is the importance of initial velocity in determining a runner's trajectory?

Initial velocity is crucial in determining a runner's trajectory as it directly affects the speed and direction at which the runner is moving. It is the starting point of the runner's motion and can greatly impact the overall trajectory.

4. Can a runner's trajectory be altered during their run?

Yes, a runner's trajectory can be altered during their run due to external factors such as wind resistance or changes in terrain. The runner's own actions, such as adjusting their speed or angle, can also impact their trajectory.

5. How does a fixed point affect a runner's trajectory?

A fixed point serves as a reference point for a runner's trajectory. The runner's aim towards this fixed point can impact their overall trajectory, as well as their ability to maintain a consistent path towards the fixed point.

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