Answer :
To solve this problem, we will follow several steps involving the conversion of units, calculation of kinetic energies, and understanding the work-energy principle. Let's break it down step-by-step.
### Step 1: Convert Speeds from km/h to m/s
First, we need to convert the given speeds from kilometers per hour (km/h) to meters per second (m/s).
- Initial speed, [tex]\( v_i \)[/tex]:
[tex]\[ v_i = 72 \, \text{km/h} = \frac{72 \times 1000}{3600} \, \text{m/s} = 20 \, \text{m/s} \][/tex]
- Final speed, [tex]\( v_f \)[/tex]:
[tex]\[ v_f = 10 \, \text{km/h} = \frac{10 \times 1000}{3600} \, \text{m/s} \approx 2.78 \, \text{m/s} \][/tex]
### Step 2: Calculate Initial and Final Kinetic Energies
The kinetic energy (KE) of an object is given by the formula:
[tex]\[ KE = \frac{1}{2} m v^2 \][/tex]
where [tex]\( m \)[/tex] is the mass and [tex]\( v \)[/tex] is the velocity.
- Initial Kinetic Energy ([tex]\( KE_i \)[/tex]):
[tex]\[ KE_i = \frac{1}{2} \times 1500 \, \text{kg} \times (20 \, \text{m/s})^2 = \frac{1}{2} \times 1500 \times 400 = 300000 \, \text{J} \][/tex]
- Final Kinetic Energy ([tex]\( KE_f \)[/tex]):
[tex]\[ KE_f = \frac{1}{2} \times 1500 \, \text{kg} \times (2.78 \, \text{m/s})^2 \approx \frac{1}{2} \times 1500 \times 7.74 \approx 5787.04 \, \text{J} \][/tex]
### Step 3: Calculate the Change in Kinetic Energy
The change in kinetic energy ([tex]\( \Delta KE \)[/tex]) is the difference between the initial and final kinetic energies:
[tex]\[ \Delta KE = KE_f - KE_i = 5787.04 \, \text{J} - 300000 \, \text{J} \approx -294212.96 \, \text{J} \][/tex]
### Step 4: Determine the Resultant Force
To find the resultant force, we need to apply the work-energy principle. The work done by the resultant force ([tex]\( F \)[/tex]) is equal to the change in kinetic energy. The formula for work done ([tex]\( W \)[/tex]) is:
[tex]\[ W = F \times d \][/tex]
where [tex]\( d \)[/tex] is the distance over which the force is applied. Since the distance [tex]\( d \)[/tex] is not given in the problem, we cannot directly calculate the exact force without this information. However, we now understand that:
- The initial kinetic energy was [tex]\( 300000 \)[/tex] J.
- The final kinetic energy was [tex]\( 5787.04 \)[/tex] J.
- The change in kinetic energy was [tex]\( -294212.96 \)[/tex] J.
This negative value indicates a decrease in the kinetic energy due to the brake force applied. However, to determine the exact force, additional information such as the distance over which the car was decelerated would be needed.
### Step 1: Convert Speeds from km/h to m/s
First, we need to convert the given speeds from kilometers per hour (km/h) to meters per second (m/s).
- Initial speed, [tex]\( v_i \)[/tex]:
[tex]\[ v_i = 72 \, \text{km/h} = \frac{72 \times 1000}{3600} \, \text{m/s} = 20 \, \text{m/s} \][/tex]
- Final speed, [tex]\( v_f \)[/tex]:
[tex]\[ v_f = 10 \, \text{km/h} = \frac{10 \times 1000}{3600} \, \text{m/s} \approx 2.78 \, \text{m/s} \][/tex]
### Step 2: Calculate Initial and Final Kinetic Energies
The kinetic energy (KE) of an object is given by the formula:
[tex]\[ KE = \frac{1}{2} m v^2 \][/tex]
where [tex]\( m \)[/tex] is the mass and [tex]\( v \)[/tex] is the velocity.
- Initial Kinetic Energy ([tex]\( KE_i \)[/tex]):
[tex]\[ KE_i = \frac{1}{2} \times 1500 \, \text{kg} \times (20 \, \text{m/s})^2 = \frac{1}{2} \times 1500 \times 400 = 300000 \, \text{J} \][/tex]
- Final Kinetic Energy ([tex]\( KE_f \)[/tex]):
[tex]\[ KE_f = \frac{1}{2} \times 1500 \, \text{kg} \times (2.78 \, \text{m/s})^2 \approx \frac{1}{2} \times 1500 \times 7.74 \approx 5787.04 \, \text{J} \][/tex]
### Step 3: Calculate the Change in Kinetic Energy
The change in kinetic energy ([tex]\( \Delta KE \)[/tex]) is the difference between the initial and final kinetic energies:
[tex]\[ \Delta KE = KE_f - KE_i = 5787.04 \, \text{J} - 300000 \, \text{J} \approx -294212.96 \, \text{J} \][/tex]
### Step 4: Determine the Resultant Force
To find the resultant force, we need to apply the work-energy principle. The work done by the resultant force ([tex]\( F \)[/tex]) is equal to the change in kinetic energy. The formula for work done ([tex]\( W \)[/tex]) is:
[tex]\[ W = F \times d \][/tex]
where [tex]\( d \)[/tex] is the distance over which the force is applied. Since the distance [tex]\( d \)[/tex] is not given in the problem, we cannot directly calculate the exact force without this information. However, we now understand that:
- The initial kinetic energy was [tex]\( 300000 \)[/tex] J.
- The final kinetic energy was [tex]\( 5787.04 \)[/tex] J.
- The change in kinetic energy was [tex]\( -294212.96 \)[/tex] J.
This negative value indicates a decrease in the kinetic energy due to the brake force applied. However, to determine the exact force, additional information such as the distance over which the car was decelerated would be needed.
