To solve this problem, we need to set up equations of motion for both you and the bus and determine when you will reach the door.

Step 1: Define the variables and equations

1. For you (running towards the bus):

   • Initial position: $x_{\text{you}}(0) = 0$ m (starting point)
   • Velocity: $v_{\text{you}} = 6.1$ m/s (constant speed)

   Your position as a function of time $t$ is given by: $x_{\text{you}}(t) = 6.1t$

2. For the bus:

   • Initial position: $x_{\text{bus}}(0) = 9$ m (starting point)
   • Initial velocity: $v_{\text{bus}} = 0$ m/s (starting from rest)
   • Acceleration: $a_{\text{bus}} = 1.0$ m/s$^2$

   The bus’s position as a function of time $t$ is given by: $x_{\text{bus}}(t) = 9 + \frac{1}{2} \cdot 1.0 \cdot t^2 = 9 + 0.5t^2$

Step 2: Set up the equation to find the time when you reach the bus

You reach the bus when your position equals the bus's position:

$6.1t = 9 + 0.5t^2$

Step 3: Rearrange the equation into a standard quadratic form

Rearrange the terms to form a quadratic equation:

$0.5t^2 - 6.1t + 9 = 0$

Multiply everything by 2 to clear the fraction:

$t^2 - 12.2t + 18 = 0$

Step 4: Solve the quadratic equation

Using the quadratic formula $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = -12.2$, and $c = 18$:

$t = \frac{-(-12.2) \pm \sqrt{(-12.2)^2 - 4 \cdot 1 \cdot 18}}{2 \cdot 1}$

$t = \frac{12.2 \pm \sqrt{148.84 - 72}}{2}$

$t = \frac{12.2 \pm \sqrt{76.84}}{2}$

$t = \frac{12.2 \pm 8.76}{2}$

Now, calculate the two possible times:

$t_1 = \frac{12.2 + 8.76}{2} = 10.48 \, \text{s}$

$t_2 = \frac{12.2 - 8.76}{2} = 1.72 \, \text{s}$

Since the only reasonable solution is the positive time when you actually reach the bus:

$\boxed{t = 1.72 \, \text{s}}$

Conclusion

It takes you approximately 1.72 seconds to reach the open door of the bus and jump in.

Would you like any further details on the solution, or do you have any questions?

Related Questions:

1. What would happen if the bus had a different acceleration?
2. How would your running speed affect the time to reach the bus?
3. What if the bus started at a higher initial velocity?
4. How does the initial distance between you and the bus impact the time required?
5. What happens if you start running a second or two after the bus starts?

Tip:

When working with equations of motion, always identify your initial conditions and variables clearly to set up accurate equations for the situation.