A car has a mass of 1500 kg. If the driver applies the brakes while on a gravel road, the maximum friction force that the tires can provide without skidding is about 7000 N.
If the car is moving at 18 m/s, what is the shortest distance in which the car can stop safely?
To determine the shortest distance in which the car can stop safely, we can use the equation of motion:
v² = u² + 2as
Where:
- v is the final velocity (0 m/s since the car stops)
- u is the initial velocity (18 m/s)
- a is the acceleration
- s is the distance
Rearranging the equation, we have:
2as = v² - u²
Since the car stops, the final velocity (v) is 0. Thus, the equation becomes:
2as = -u²
Plugging in the given values:
- u = 18 m/s
2as = -(18 m/s)²
Next, we need to find the acceleration (a). The maximum friction force that the tires can provide without skidding is equal to the product of the mass (m) and the acceleration (a):
F_fric = m * a
Rearranging the equation, we can solve for a:
a = F_fric / m
Plugging in the given values:
- F_fric = 7000 N
- m = 1500 kg
a = 7000 N / 1500 kg
Now, we can substitute the calculated acceleration (a) into the equation:
2as = -(18 m/s)²
2 * (7000 N / 1500 kg) * s = -(18 m/s)²
Simplifying further:
(2 * 7000 N * s) / 1500 kg = -(18 m/s)²
Multiplying both sides by 1500 kg:
2 * 7000 N * s = -1500 kg * (18 m/s)²
Multiplying both sides by 2:
14000 N * s = -1500 kg * 324 m²/s²
Dividing both sides by 14000 N:
s = (-1500 kg * 324 m²/s²) / 14000 N
Evaluating the expression:
s ≈ -54 m²/s²
Since we are looking for a positive distance, we take the absolute value:
s ≈ | -54 | = 54 m
Therefore, the shortest distance in which the car can stop safely is approximately 54 meters.
To find the shortest distance in which the car can stop safely, we need to determine the deceleration of the car first. We can use the equation:
F = m * a
Where:
F is the force of friction
m is the mass of the car
a is the acceleration (deceleration in this case)
Given:
m = 1500 kg
F = 7000 N
Rearranging the equation, we can solve for a:
a = F / m
Substituting the values:
a = 7000 N / 1500 kg
a ≈ 4.67 m/s²
Now, we can use the formula to calculate the stopping distance:
v² = u² - 2a * s
Where:
v is the final velocity (0 m/s, since the car stops)
u is the initial velocity (18 m/s)
a is the deceleration (4.67 m/s²)
s is the stopping distance (what we are trying to find)
Rearranging the equation, we can solve for s:
s = (u² - v²) / (2a)
Substituting the values:
s = (18 m/s)² / (2 * 4.67 m/s²)
s ≈ 17.79 m
Therefore, the shortest distance in which the car can stop safely is approximately 17.79 meters.