Question

You are driving your car along a country road at a speed of $27.0 \mathrm{m} / \mathrm{s} .$ As you come over the crest of a hill, you notice a farm tractor \(25.0 \mathrm{m}\) ahead of you on the road, moving in the same direction as you at a speed of \(10.0 \mathrm{m} / \mathrm{s} .\) You immediately slam on your brakes and slow down with a constant acceleration of magnitude $7.00 \mathrm{m} / \mathrm{s}^{2} .$ Will you hit the tractor before you stop? How far will you travel before you stop or collide with the tractor? If you stop, how far is the tractor in front of you when you finally stop?

Short answer

If not, how far apart will they be when the car comes to a complete stop? Answer: The car will not hit the tractor. When the car comes to a complete stop, it will be 11.58 meters away from the tractor.

Step-by-step solution

Calculate time needed for the car to stop

We'll use the following equation of motion as the car slows down with constant acceleration: \(v = u + at\) Where, \(v\) is the final velocity, which is \(0 \mathrm{m/s}\) because the car stops \(u\) is the initial velocity of the car \(27.0 \mathrm{m/s}\) \(a\) is the acceleration of the car \(-7.00 \mathrm{m/s^2}\) (note that it's negative because it's decelerating) \(t\) is the time we are trying to calculate. Solve for \(t\): \(t = \frac{v - u}{a}\) Now, calculate the time: \(t = \frac{0 - 27.0}{-7.00} = 3.86 \mathrm{s}\) Next, we'll find the distance covered by the car during this time:

Calculate the distance traveled by the car until it stops

We'll use this equation of motion: \(s = ut + \frac{1}{2}at^2\) Where, \(s\) is the distance traveled by the car \(u\), \(a\), and \(t\) have the same values as in the previous step Now, calculate the distance: \(s = 27.0 * 3.86 + \frac{1}{2} * (-7.00) * (3.86)^2 = 52.02 \mathrm{m}\) Next, let's find the distance covered by the tractor during the same time:

Calculate the distance traveled by the tractor

We'll use the basic distance equation for the tractor since it has no acceleration: \(s_\mathrm{tractor} = u_\mathrm{tractor} * t\) Where, \(s_\mathrm{tractor}\) is the distance traveled by the tractor \(u_\mathrm{tractor} = 10.0 \mathrm{m/s}\) is the initial velocity of the tractor \(t = 3.86 \mathrm{s}\) is the time we found in step 1. Now, calculate the distance: \(s_\mathrm{tractor} = 10.0 * 3.86 = 38.6 \mathrm{m}\) Now, let's determine if the car will hit the tractor or not and calculate the distance between them if the car stops before hitting the tractor.

Check if the car will collide with the tractor and calculate the final distance

Compare the distance traveled by the car and the distance between the car and the tractor plus the distance traveled by the tractor: \(s > 25.0 + s_\mathrm{tractor}\) ➞ \(52.02 > 25.0 + 38.6\) ➞ \(52.02 > 63.6\) The inequality is false, so the car won't hit the tractor. To find the final distance between the car and the tractor, we will subtract the distance that the car traveled from the initial distance between them plus the distance the tractor traveled: \(Distance_\mathrm{final} = (25.0 + s_\mathrm{tractor}) - s\) Calculate the final distance: \(Distance_\mathrm{final} = (25.0 + 38.6) - 52.02 = 11.58 \mathrm{m}\) So, the car won't hit the tractor, and when the car comes to a complete stop, it will be 11.58 meters away from the tractor.