Question
A train of mass 100000000kg is Travelling at 2ms^-1 when its engine is switched off as a consequence the ships speed is reduced to 1.5ms^-1at a distance of 100m. Assuming the resistance to the ships motion is uniform, Calculate the magnitude of the resistance
Answers
GPT-4o mini
To calculate the magnitude of the resistance acting on the train, we can use the work-energy principle. The work done by the resistance force is equal to the change in kinetic energy of the train.
1. **Calculate the initial kinetic energy (KE_initial) when the train is traveling at 2 m/s:**
\[
\text{KE}_{\text{initial}} = \frac{1}{2} m v^2
\]
Where:
- \( m = 100,000,000 \, \text{kg} \) (mass of the train)
- \( v = 2 \, \text{m/s} \) (initial speed)
\[
\text{KE}_{\text{initial}} = \frac{1}{2} \times 100,000,000 \times (2)^2 = \frac{1}{2} \times 100,000,000 \times 4 = 200,000,000 \, \text{J}
\]
2. **Calculate the final kinetic energy (KE_final) when the train is traveling at 1.5 m/s:**
\[
\text{KE}_{\text{final}} = \frac{1}{2} m v^2
\]
Where:
- \( v = 1.5 \, \text{m/s} \) (final speed)
\[
\text{KE}_{\text{final}} = \frac{1}{2} \times 100,000,000 \times (1.5)^2 = \frac{1}{2} \times 100,000,000 \times 2.25 = 112,500,000 \, \text{J}
\]
3. **Calculate the change in kinetic energy (ΔKE):**
\[
\Delta KE = \text{KE}_{\text{final}} - \text{KE}_{\text{initial}}
\]
\[
\Delta KE = 112,500,000 - 200,000,000 = -87,500,000 \, \text{J}
\]
4. **Calculate the work done by the resistance force (W_resistance):**
By the work-energy principle, the work done by the resistance force is equal to the change in kinetic energy.
\[
W_{\text{resistance}} = -\Delta KE = 87,500,000 \, \text{J}
\]
5. **Calculate the resistance force (R):**
The work done by the resistance force can also be expressed as:
\[
W_{\text{resistance}} = R \times d
\]
Where \( d = 100 \, \text{m} \) (distance over which resistance acts).
Rearranging for \( R \):
\[
R = \frac{W_{\text{resistance}}}{d} = \frac{87,500,000}{100}
\]
\[
R = 875,000 \, \text{N}
\]
Thus, the magnitude of the resistance is \( \boxed{875000 \, \text{N}} \).
1. **Calculate the initial kinetic energy (KE_initial) when the train is traveling at 2 m/s:**
\[
\text{KE}_{\text{initial}} = \frac{1}{2} m v^2
\]
Where:
- \( m = 100,000,000 \, \text{kg} \) (mass of the train)
- \( v = 2 \, \text{m/s} \) (initial speed)
\[
\text{KE}_{\text{initial}} = \frac{1}{2} \times 100,000,000 \times (2)^2 = \frac{1}{2} \times 100,000,000 \times 4 = 200,000,000 \, \text{J}
\]
2. **Calculate the final kinetic energy (KE_final) when the train is traveling at 1.5 m/s:**
\[
\text{KE}_{\text{final}} = \frac{1}{2} m v^2
\]
Where:
- \( v = 1.5 \, \text{m/s} \) (final speed)
\[
\text{KE}_{\text{final}} = \frac{1}{2} \times 100,000,000 \times (1.5)^2 = \frac{1}{2} \times 100,000,000 \times 2.25 = 112,500,000 \, \text{J}
\]
3. **Calculate the change in kinetic energy (ΔKE):**
\[
\Delta KE = \text{KE}_{\text{final}} - \text{KE}_{\text{initial}}
\]
\[
\Delta KE = 112,500,000 - 200,000,000 = -87,500,000 \, \text{J}
\]
4. **Calculate the work done by the resistance force (W_resistance):**
By the work-energy principle, the work done by the resistance force is equal to the change in kinetic energy.
\[
W_{\text{resistance}} = -\Delta KE = 87,500,000 \, \text{J}
\]
5. **Calculate the resistance force (R):**
The work done by the resistance force can also be expressed as:
\[
W_{\text{resistance}} = R \times d
\]
Where \( d = 100 \, \text{m} \) (distance over which resistance acts).
Rearranging for \( R \):
\[
R = \frac{W_{\text{resistance}}}{d} = \frac{87,500,000}{100}
\]
\[
R = 875,000 \, \text{N}
\]
Thus, the magnitude of the resistance is \( \boxed{875000 \, \text{N}} \).