Rocket#356

Rocket (Editorial)

A rocket is launched with an initial velocity $V$ and constant acceleration $A$ . Our goal is to find the first meter along the rocket's path that it will traverse in less than a given time $T$ .

As the rocket accelerates, each subsequent meter will take less time to travel.

Example

Suppose the inputs are:

Acceleration $A = 2 m/s^{2}$ A = 2 \, \text{m/s}^2
Initial velocity $V = 1 m/s$ V = 1 \, \text{m/s}
Target time $T = 1 s$ T = 1 \, \text{s}

To solve this, we calculate the time taken to travel each meter until we find the first one completed in under $T$ seconds.

Kinematic Equation with Initial Velocity

We use the kinematic equation for motion with constant acceleration:

S = V \times T + \frac{1}{2} A \times T^{2}

S = V \times T + \frac{1}{2} A \times T^2

where:

$S$ is the distance traveled.
$V$ is the initial velocity.
$A$ is the constant acceleration.
$T$ is the time taken.

To find the time $T$ to reach a given distance $S$ , we rearrange the equation as:

T = \frac{- V + V ^{2} + 2 \cdot A \cdot S}{A}

T = \frac{-V + \sqrt{V^2 + 2 \cdot A \cdot S}}{A}

This formula provides the time needed to travel a distance $S$ from the starting point.

Calculating the Time to Traverse Each Meter

To find out how long it takes to traverse each individual meter:

Calculate the time $t_{n - 1}$ t_{n-1} to reach the position at $(N - 1)$ (N-1) meters.
Calculate the time $t_{n}$ t_n to reach $N$ meters.
The time taken to traverse the $N$ -th meter alone is:
$Δ t_{N} = t_{N} - t_{N - 1}$
\Delta t_N = t_N - t_{N-1}

Writing the Program

We implement this step-by-step in C++ as follows:

Set up variables for acceleration, initial velocity, and target time. Initialize a res variable to 1, representing the current meter.
Calculate $t_{N}$ t_N and $t_{N - 1}$ t_{N-1} for each meter incrementally.
Check if $Δ t_{N}$ \Delta t_N is less than $T$ . If so, we've found our answer. Otherwise, increment the meter and continue.

Full Code Example

int main() {
    double A, V, T, time=0.0, now, delta;
    int res = 1;
    cin >> A >> V >> T;
    
    while (true) {
        // Calculate time to reach the end of the current meter
        //        using formula mentioned before
        now = (-V + sqrt(V * V + 2 * A * res)) / A;
        // Time taken to traverse this meter
        delta = now - time;
        // Check if time to traverse this meter is less than target time
        if (delta < T) {
            cout << res << endl;
            return 0;
        }
        time = now;
        res++;
    }
}

C++