When a high speed passenger train travelling at 161km/h rounds a bend,the engineer is shocked to see that a locomtive has improperly entered onto track from a siding and is a distance D=676m ahead.The locomotive is moving at 29km/h. The engineer of the high speed train immediately applies the brakes. What must be the magnitude of the resulting constant decelaration if a collision is to be just avoided?(note: both the train and the locomotive are moving in the same direction)
If the trains barely avoid collision during a decelerating period T, they both end up at the same place at that time. Let's measure X from the position of the faster train when deceleration starts. Let X1 be the distance travelled by the faster train, and X2 bew the distance travelled by the other train, at time T.
X1 = 161 T - (1/2) a T^2
X2 = 676 + 29 T
Set X1 = X2 and you still have two unknowns, T and a. You have one other equation to work with:
a T = 161
which says that the velocity of the faster train becomes zero at t = T.
Eliminate the variable T and solve for a
Thanks for the reply. But, i don't quite get where do u get X2 equation from.
To solve this problem, we can first calculate the time it would take for the high-speed train to reach the location of the locomotive if it maintained its initial velocity. Then, we can calculate the distance the train would travel during that time. If the train can stop before reaching that distance, a collision will be avoided.
Let's break down the problem and calculate the necessary values step by step.
Step 1: Convert the velocities to m/s
The velocity of the high-speed train is given as 161 km/h. To convert this to m/s, we divide by 3.6:
Velocity of the high-speed train = 161 km/h ÷ 3.6 = 44.7 m/s
The velocity of the locomotive is given as 29 km/h. Converting this to m/s:
Velocity of the locomotive = 29 km/h ÷ 3.6 = 8.1 m/s
Step 2: Calculate the time it takes for the train to reach the locomotive
The relative velocity between the train and the locomotive is the difference between their velocities:
Relative velocity = Velocity of the high-speed train - Velocity of the locomotive
= 44.7 m/s - 8.1 m/s
= 36.6 m/s
To calculate the time it takes for the train to reach the locomotive, we can use the formula:
Time = Distance / Velocity
Time = 676 m / 36.6 m/s
Time ≈ 18.46 s
Step 3: Calculate the distance the train would travel during that time
Using the formula:
Distance = Velocity × Time
Distance = 44.7 m/s × 18.46 s
Distance ≈ 824.56 m
Step 4: Calculate the necessary deceleration to avoid a collision
To avoid a collision, the train must stop before reaching the distance traveled during the time it takes to reach the locomotive. So, the necessary deceleration can be calculated using the formula for uniform deceleration:
Distance = (Initial Velocity × Time) + (0.5 × Deceleration × Time²)
Rearranging the formula to solve for deceleration:
Deceleration = (Distance - (Initial Velocity × Time)) / (0.5 × Time²)
Deceleration = (824.56 m - (44.7 m/s × 18.46 s)) / (0.5 × (18.46 s)²)
Calculating this, we find:
Deceleration ≈ 7.97 m/s²
Therefore, the magnitude of the resulting constant deceleration must be approximately 7.97 m/s² in order to avoid a collision.