Problems dealing with motion are usually the first that students of physics will encounter. Concepts like time, velocity and acceleration are interrelated by formulas that students can rearrange with the help of algebra to apply to different circumstances. Students can calculate the height of a jump, for instance, from more than one starting point. If you know the acceleration and either the initial velocity or the total time in the air, you can calculate the height of the jump.
Derive the formula vi = -g_T/2 from the formula sf = si + vi_T + (g_T²)/2. The initial and final positions are the same before and after the jump, so set them to zero and factor: T(vi + g_T/2) = 0. Setting the factors equal to zero gives you two results: T = 0 and vi + g*T/2 = 0. The first indicates that no time is required for your initial position to equal your final, and the second result represents the amount of time for a body to rise and fall back down to where it was. Solve the second expression for initial velocity.
Write an expression for time in terms of change in velocity, using the formula vf = g*t + vi, where vf is final velocity, g is the acceleration due to gravity, t is time, and vi is initial velocity. Solve the equation for t: t = (vf - vi)/g. Therefore, the amount of time is equal to the change in velocity divided by the acceleration due to gravity.
Calculate the amount of time to reach the highest point of the jump. At the highest point, velocity is zero, so with the initial velocity and the formula t = (vf - vi)/g, you can find time. Use -9.8 meters/second² for the acceleration due to gravity. For example, if the initial velocity is 1.37 meters/second, time is:
t = (0 - 1.37)/(-9.8) t = 0.14 seconds
If you know the total time in the air, you can calculate the initial velocity with the formula vi = -g*T/2, where T is total time. Total time is also twice the time to reach the highest point, so t = T/2. For example, if the total time is 0.28 seconds:
vi = -(-9.8 * 0.28)/2 vi = 1.37 meters per second t = 0.28/2 t = 0.14 seconds
Calculate the jump height using the formula sf = si + vi_t + (g_t²)/2, where sf is the final position and si is the initial position. Since jump height is the difference between the final and initial position, h = (sf - si), simplify the formula to h = vi_t + (g_t²)/2, and calculate:
h = (1.37_0.14) + (-9.8_0.14²)/2 h = 0.19 - 0.10 h = 0.09 meters
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