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A particle moving in a straight line was accelerating at a rate of 22 centimeters per square second in the same direction as its initial velocity.
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If the magnitude of its displacement 10 seconds after it started moving was 29 meters, calculate the magnitude of its initial velocity 𝑣 naught and its velocity 𝑣 at the end of this period.
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We’re given that the particle is accelerating at a constant rate of 22 centimeters per square second.
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This means to answer this question, we’re going to need to use the kinematic equations.
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These are, of course, equations of constant acceleration.
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For a starting velocity 𝑣 naught, a velocity 𝑣 after 𝑡 time units, an acceleration 𝑎, and a displacement Δ𝑥, they are as shown.
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What we do is begin by listing everything we know about our motion.
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We’ve already said we know that the acceleration is constant, and it’s 22 centimeters per square second.
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It’s in the same direction as its initial velocity.
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Now, we don’t know its initial velocity, but by assuming that they’re in the same direction, we can take both acceleration and 𝑣 naught to be positive.
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We’re also told that the magnitude of the displacement 10 seconds after it started moving was 29 meters.
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Remember, displacement can have a direction.
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So, by considering just the magnitude, we’re thinking about the distance; that’s 29 meters.
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Time 𝑡 is 10 seconds.
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Now, the question actually asks us to calculate the magnitude of the initial velocity and its velocity at the end of the period.
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Let’s begin by calculating its initial velocity 𝑣 naught.
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In this case, we’re not interested in 𝑣, so we go through our equations and eliminate those containing 𝑣.
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Those are one, three, and four.
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Our next step would normally be to substitute everything we know about the motion of our particle into that second equation.
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We do have a little bit of a problem though.
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We notice that the units for our acceleration and our displacement are different.
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We need them to be the same.
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So, we multiply displacement by 100 and we find it’s actually equal to 2900 centimeters.
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Then, substituting everything we know into this formula, and we get 2900 equals 10𝑣 naught plus a half times 22 times 10 squared.
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A half times 22 times 10 squared is 1100.
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So, we subtract 1100 from both sides, and we find that 1800 is equal to 10 times 𝑣 naught.
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Our final step is to divide through by 10.
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1800 divided by 10 is 180.
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Now, we’re working in centimeters.
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So, our velocity, our initial velocity 𝑣 naught, is 180 centimeters per second.
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We might choose to give our answer in meters per second by dividing through by 100.
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And when we do, we find that 𝑣 naught is 1.8 meters per second.
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We’re not quite finished.
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We’re still looking to calculate its velocity 𝑣 at the end of the period.
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Now that we know 𝑣 naught, we can actually use any of our equations.
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So, let’s use the first one.
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We substitute everything we know about the motion of our particle into this formula, continuing to work in centimeters and centimeters per second.
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When we do, we get 𝑣 is 180 plus 22 times 10.
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22 times 10 is 220.
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And 180 plus 220 is 400.
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We’re still, of course, working in centimeters per second.
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To give our answer in meters per second, we’ll divide through by 100.
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And when we do, we find that the velocity 𝑣 at the end of the motion is four meters per second.