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BY OUR STUDENT CONTRIBUTORS
Malhar Rajpal If you’re a high school physics student, you’ve probably encountered the 5 SUVAT equations. These are equations of motion (kinematics equations) that are used when acceleration is constant. Most students memorise and accept the equations without knowing where they came from. In this article, I will derive three of the SUVAT equations using basic calculus and explain why they work. Above are the 5 SUVAT equations where s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time. Note that s, u , v, and a are all vector quantities, so they have both direction and magnitude. In this context, that means a positive quantity represents motion in one direction (e.g. upwards), and a negative quantity represents motion in the opposite direction (e.g. downwards). Proof 1: v = u + at We know acceleration is the rate of change of velocity, v, so we can say a=dv/dt. We can multiply both sides by dt to get dv = a dt. Integrating this gives the following: When t = 0, at equals 0 and v = C. At time 0, v is the same as the initial velocity, which is u. We can therefore say that C = u, so our equation becomes v = at + u, as required. Proof 2: s = ut + (1/2)a(t^2) We know that velocity is the rate of change of displacement, s, so we can say v=ds/dt. We can multiply both sides by dt to get ds = v dt. Integrating this gives the following: We know from equation 1 that v = u + at, so substituting this in gives: When t = 0, s = C. At time 0, displacement is also 0, since the object has not had any time to move, so C = 0. Our equation then becomes s = ut + (1/2)a(t^2), as required. Proof 3: v^2 = u^2 + 2as From the first SUVAT equation, we know that v = u + at. Squaring both sides and rearranging gives the following: Do you see something familiar? Yes! ut + (1/2)a(t^2) the right hand side of the second SUVAT equation! By using that equation, we can replace ut + (1/2)a(t^2) with s and rewrite the initial equation as v^2 = u^2 + 2as , the third SUVAT equation!
Now that you have seen how to derive the first three SUVAT equations, I want to leave it to you to try and derive the last two SUVAT equations using either calculus or the other SUVAT equations! Don’t worry if you get stuck, because my next article will be on proving the fourth and fifth SUVAT equations!
2 Comments
Greg Menkiti
26/4/2022 05:25:06 am
Thank you
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17/3/2023 08:19:16 pm
i love this article. helped me recover rom my severely melancholic mood. also join my website to see more of my content. <3
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