Calculate the force needed to bring a 1100–kg car to rest from a speed of 83.0 km/h in a distance of 118 m (a fairly typical distance for a non-panic stop). Suppose instead the car hits a concrete abutment at full speed and is brought to a stop in 1.82 m. Calculate the force exerted on the car and compare it with the force found in part (a).
In this case work takes away KE so
1/2 mv^2 = Fd (convert kph to m/s BTW)
Solve for F in both questions
To calculate the force needed to bring the car to rest, we can use the equation:
Force = (mass * change in velocity) / time
Here, we are given the mass of the car (1100 kg) and the initial velocity (83.0 km/h). We need to convert the velocity to meters per second (m/s) before using it in the equation.
Converting km/h to m/s:
1 km/h = 1000 m/3600 s
83.0 km/h = (83.0 * 1000) / 3600 = 23.06 m/s (approx.)
Now, we also need to find the change in velocity. Since the car is brought to rest, the final velocity is 0 m/s. Thus, the change in velocity is:
Change in velocity = 0 m/s - 23.06 m/s = -23.06 m/s
We are not given the time taken to stop in this case, but we are given a distance of 118 m. Assuming the car is brought to rest at a constant deceleration, we can use the following equation to find the time taken:
Distance = (initial velocity * time) + (0.5 * acceleration * time^2)
Here, the initial velocity is 23.06 m/s, the distance is 118 m, and we assume the final velocity is 0 m/s. Solving this equation for time:
118 = (23.06 * t) + (0.5 * a * t^2) ---(1)
Since the car is brought to rest, the final velocity is 0 m/s. Rearranging equation (1) gives us:
0.5 * a * t^2 + 23.06 * t - 118 = 0
This equation can be solved for time (t) using the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac)) / (2a)
where a = 0.5, b = 23.06, and c = -118.
Solving this equation with the positive root will give us the time taken to stop.
Once we find the time, we can substitute the mass (1100 kg), change in velocity (-23.06 m/s), and time (from previous step) into the equation to calculate the force required to bring the car to rest.
To find the force exerted on the car when it hits a concrete abutment at full speed and is brought to a stop in 1.82 m, we can use the same equation:
Force = (mass * change in velocity) / time
However, in this case, we are given the distance (1.82 m) instead of the time. We can assume the car comes to a stop with constant deceleration, using the equation used before:
Distance = (initial velocity * time) + (0.5 * acceleration * time^2)
In this case, the initial velocity is 23.06 m/s, the distance is 1.82 m, and we assume the final velocity is 0 m/s. Rearranging this equation will give us the time taken to stop.
Once we find the time, we can substitute the mass (1100 kg), change in velocity (-23.06 m/s), and time (from previous step) into the equation to calculate the force exerted on the car.
Finally, we can compare the forces calculated in both parts to determine how they differ.