Speedy Sue, driving at 30.0 m/s enters a one-lane tunnel. She then observes a slow-moving van 155m ahead traveling at 5.00 m/s. Sue applies her brakes but can accelerate only at -2.00m/s/s because the road is wet. Will there be a collision? If yes, determine how far into the tunnel and at what time the collision occurs.
To determine if there will be a collision, we need to calculate the time it takes for Speedy Sue to reach the slow-moving van. We can do this by dividing the distance between Sue and the van by Sue's relative velocity.
The relative velocity can be found by subtracting the van's velocity from Sue's velocity:
Relative velocity = Sue's velocity - Van's velocity = 30.0 m/s - 5.00 m/s = 25.0 m/s
Next, we calculate the time it takes for Sue to reach the van using the formula:
Time = Distance / Velocity
Time = 155 m / 25.0 m/s = 6.20 seconds
Now, let's determine if there will be a collision. To do this, we need to determine whether Sue can stop before reaching the van within the given acceleration.
We can calculate the distance Sue needs to decelerate from her initial velocity to a complete stop using the formula:
Distance = (Velocity^2 - Final Velocity^2) / (2 * Acceleration)
Where:
- Velocity is Sue's initial velocity = 30.0 m/s
- Final Velocity is the velocity at which Sue stops = 0 m/s
- Acceleration is given as -2.00 m/s^2 (since it is decelerating on the wet road)
Distance = (30.0 m/s)^2 - (0 m/s)^2 / (2 * -2.00 m/s^2) = -450 m
The negative sign indicates that the distance required for Sue to stop is greater than the initial distance of 155 m. Therefore, Sue cannot stop in time, and there will be a collision.
To calculate how far into the tunnel the collision occurs, we subtract the distance Sue can travel before stopping from the initial distance between Sue and the van:
Collision Distance = Initial Distance - Distance Sue can travel before stopping
Collision Distance = 155 m - (-450 m) = 605 m
Therefore, the collision occurs 605 meters into the tunnel.
To determine the time at which the collision occurs, we multiply the time it takes for Sue to reach the van by the proportion of how close Sue is to the van when she applies the brakes.
Proportion of distance traveled = Distance Sue can travel before stopping / Initial Distance
Proportion of distance traveled = (-450 m) / 155 m = -2.903
Collision Time = Time to reach the van * Proportion of distance traveled
Collision Time = 6.20 s * -2.903 = -18 s
The negative sign indicates that the collision occurs 18 seconds before Sue reached the van.
Therefore, the collision occurs 605 meters into the tunnel, and it happens 18 seconds before Sue reaches the slow-moving van.