A 2400-pound car exerts its maximum force of 750 pounds. What will be its minimum time needed to slow it down from 50 m/s to 10 m/s? Determine how far will the car travel in that time.
deceleration rate =
a = (velocity change)/time
= -F/m = -750 lb/74.5 slugs
= -10.06 ft/s^2
Time = (velocity change)/a
Velocity change must be in ft/s units if you are going to use pounds for mass and slugs for force.
delta V = -40 m/s = -131.2 ft/s
Time = (-131.2 ft/s)/(-10.06 ft/s^2)
= 13.0 sec
Multiply that by the average velocity (30 m/s) for the distance (in meters)
To determine the minimum time needed to slow the car down from 50 m/s to 10 m/s, we can use the equation:
F = ma
where F is the force applied, m is the mass of the car, and a is the acceleration.
Given that the force exerted is 750 pounds and the car weighs 2400 pounds, we need to convert the weights to mass by dividing them by the acceleration due to gravity (32.2 ft/s^2). So,
m = 2400 lbs / 32.2 ft/s^2 = 74.53 slugs
F = 750 lbs / 32.2 ft/s^2 = 23.29 slugs * ft/s^2 (force is also expressed as lb in * ft/s^2)
Now, let's calculate the acceleration:
F = ma
23.29 slugs * ft/s^2 = 74.53 slugs * a
a ≈ 0.312 ft/s^2
To determine the time required, we can use the equation:
v = u + at
where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
Given that the initial velocity (u) is 50 m/s, the final velocity (v) is 10 m/s, and the acceleration (a) is 0.312 ft/s^2, we need to convert the velocities to feet per second:
50 m/s ≈ 164.04 ft/s
10 m/s ≈ 32.81 ft/s
Substituting the values into the equation:
32.81 ft/s = 164.04 ft/s + (0.312 ft/s^2) * t
Simplifying the equation:
-131.23 ft/s = (0.312 ft/s^2) * t
Dividing both sides by 0.312 ft/s^2:
t ≈ -415.1 s
Since time cannot be negative, there seems to be an error in the calculation or the given values. Could you please double-check the provided data?
To find the minimum time needed to slow down the car and the distance it will travel in that time, we can use the equations of motion.
Let's start by finding the acceleration of the car using the force exerted and its mass.
Step 1: Convert the car's mass to kilograms.
1 pound is equal to approximately 0.4536 kilograms. Therefore, the car's mass in kilograms will be:
2400 pounds * 0.4536 kg/pound = 1088.64 kg (rounded to two decimal places).
Step 2: Calculate the acceleration of the car.
Using Newton's second law of motion, force = mass * acceleration, we can rearrange the equation to find acceleration:
acceleration = force / mass = 750 pounds / 1088.64 kg = 0.689 m/s^2 (rounded to three decimal places).
Now that we have the acceleration of the car, we can proceed to find the minimum time needed to slow down and the distance traveled.
Step 3: Calculate the change in velocity (Δv).
The change in velocity is the difference between the initial velocity and the final velocity:
Δv = 50 m/s - 10 m/s = 40 m/s.
Step 4: Calculate the time (t) using the kinematic equation:
Δv = acceleration * t
t = Δv / acceleration = 40 m/s / 0.689 m/s^2 ≈ 58.03 s (rounded to two decimal places).
Therefore, the minimum time needed to slow down the car from 50 m/s to 10 m/s is approximately 58.03 seconds.
Step 5: Calculate the distance (s) traveled during that time.
Using the equation of motion: s = ut + 0.5at^2, where u is the initial velocity.
Since the car starts at 50 m/s and the acceleration is constant at 0.689 m/s^2, we can calculate the distance using:
s = 50 m/s * 58.03 s + 0.5 * 0.689 m/s^2 * (58.03 s)^2 ≈ 3105.91 meters (rounded to two decimal places).
Therefore, the car will travel approximately 3105.91 meters during the minimum time needed to slow down from 50 m/s to 10 m/s.