Okay, let's tackle this problem step by step. So, the question is about the ratio of accelerations when two different balls are thrown with the same force. Hmm, right. Let me recall the basic physics here. The formula given is force equals mass times acceleration, which is F = m * a. 

So, Jen is throwing a softball with a force equal to Jack's baseball throw. The softball has a mass of 200 grams, and the baseball is 150 grams. We need to find the ratio of Jen's acceleration to Jack's acceleration. 

First, let me make sure I understand the problem correctly. Both Jen and Jack apply the same force when they throw their respective balls. The softball is heavier than the baseball, so according to F = ma, since the force is the same, the acceleration should be inversely proportional to the mass. That makes sense because a heavier object accelerates less when the same force is applied. 

So, if I denote the force as F, then for Jack throwing the baseball: F = m_jack * a_jack. For Jen throwing the softball: F = m_jen * a_jen. Since the forces are equal, we can set these two equations equal to each other: m_jack * a_jack = m_jen * a_jen. 

We need to find the ratio of acceleration of Jen's ball to Jack's, which is a_jen / a_jack. Rearranging the equation above, we can solve for a_jen / a_jack. Let's see:

From m_jack * a_jack = m_jen * a_jen, divide both sides by m_jack and a_jen:

a_jen / a_jack = m_jack / m_jen.

Wait, is that right? Let me check. If we divide both sides by m_jack * a_jen, then we get a_jen / a_jack = m_jack / m_jen. Yes, that seems correct. So the ratio of accelerations is the ratio of the masses of the baseball to the softball.

But wait, the mass of the baseball is 150 grams, and the softball is 200 grams. So m_jack is 150 g, m_jen is 200 g. Therefore, the ratio a_jen / a_jack is 150 / 200. Simplifying that fraction, divide numerator and denominator by 50: 150 ÷ 50 = 3, 200 ÷ 50 = 4. So 3/4. Therefore, the ratio is 3/4.

But let me make sure I didn't mix up the ratio. The question asks for the ratio of Jen's acceleration to Jack's. Jen is the one throwing the softball, which is heavier. So since a heavier object accelerates less, Jen's acceleration should be less than Jack's. Therefore, the ratio a_jen / a_jack should be less than 1, which 3/4 is. That makes sense.

Alternatively, if I write the equation again: F = m_j * a_j = m_k * a_k. So a_j / a_k = m_k / m_j. So yes, the ratio is the mass of the other object divided by their own mass. So in this case, since Jen's ball (softball) is m_j = 200g, and Jack's is m_k = 150g, then a_j / a_k = 150 / 200 = 3/4.

Therefore, the answer should be 3/4. Let me just verify once more. If Jack's ball has less mass, same force, higher acceleration. So Jen's ball (200g) has more mass, so less acceleration. So the ratio is 3/4. Yep, that seems right.

I don't think I made any mistakes here. The key was recognizing that with the same force, acceleration is inversely proportional to mass. So heavier object has smaller acceleration. Therefore, the ratio is 150/200 simplified to 3/4.
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