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Chapter 7: Problem 13

An automobile traveling at a speed of \(30.0 \mathrm{m} / \mathrm{s}\) applies its brakes and comes to a stop in 5.0 s. If the automobile has a mass of $1.0 \times 10^{3} \mathrm{kg},$ what is the average horizontal force exerted on it during braking? Assume the road is level.

Short Answer

Expert verified
Answer: The average horizontal force exerted on the automobile during braking is \(-6.0 \times 10^3\) N in the opposite direction of the initial velocity.

Step by step solution

01

Identify the given information

From the problem statement, we are given the following information: - initial speed = 30.0 m/s - final speed = 0 m/s (since the automobile comes to a stop) - time = 5.0 s - mass = \(1.0 \times 10^3 kg\)
02

Calculate the acceleration

We need to find the acceleration of the automobile to use Newton's second law of motion. We can do this by using the equation for the acceleration of an object: $$a = \frac{v_f - v_i}{t}$$ Where \(a\) is the acceleration, \(v_i\) is the initial speed, \(v_f\) is the final speed, and \(t\) is the time. Plugging in the given values, we have: $$a = \frac{0 \,\text{m/s} - 30.0 \,\text{m/s}}{5.0 \,\text{s}}$$
03

Simplify the acceleration equation

Now, we need to simplify the equation to find the acceleration: $$a = \frac{-30.0 \,\text{m/s}}{5.0 \,\text{s}} = -6.0 \, \mathrm{m/s^2}$$ The negative sign indicates that the acceleration is in the opposite direction of the initial velocity, which makes sense since the automobile is slowing down.
04

Use Newton's second law to find the force

Now that we have the acceleration, we can use Newton's second law of motion to find the average horizontal force exerted on the automobile during braking: $$F = ma$$ Where \(F\) is the force, \(m\) is the mass, and \(a\) is the acceleration. Plugging in the given mass and calculated acceleration, we have: $$F = (1.0 \times 10^3 \,\text{kg})(-6.0 \, \mathrm{m/s^2})$$
05

Calculate the force

Finally, we can simplify the equation to find the force: $$F = -6.0 \times 10^3 \, \text{N}$$ The negative sign indicates that the force is in the opposite direction of the initial velocity, which is expected since the force is acting to slow down the automobile.
06

State the final answer

The average horizontal force exerted on the automobile during braking is \(-6.0 \times 10^3\) N in the opposite direction of the initial velocity.

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