Equations of Motion PDF

Equations of Motion Objective In this lesson, you will solve problems for objects with constant acceleration, relating displacement, velocity, acceleration, and time. Equations of Kinematics velocity Acceleration is the chan...

Equations of Motion Objective In this lesson, you will solve problems for objects with constant acceleration, relating displacement, velocity, acceleration, and time. Equations of Kinematics velocity Acceleration is the change in ________________ of an object in a given time interval. We can use an equation to find the average acceleration of an object in one dimension over a given time period: ∆𝒗 𝒗 − 𝒗𝟎 𝒂= = , where ∆𝒕 𝒕 final v = ____________ velocity (meters/second), initial v0 = _____________ velocity (meters/second), a = acceleration (meters/second2), and time t = Δt = __________ (seconds). accelerated Rearranging this equation, we get the velocity equation for ___________________ motion in one dimension: v = v0 + at. constant Let's look at an example of velocity under ______________ acceleration. Suppose a car moving on a horizontal road has an initial velocity of +10 meters/second and a constant acceleration of 2 meters/second2. What will its final velocity be after 5 seconds? We're looking for final velocity, v. We can use the velocity equation, substituting in the known values: v = v0 + at 2 = +10 m/s + (_____ 5 m/s2)(_____ s) 10 = +10 m/s + (______ m/s) 20 = ______ m/s 20 The final velocity of the car after 5 seconds is ______ meters/second. gravity The force of ________________ is our most familiar source of constant acceleration. Displacement Equation In a car race, each racer drives the car from its original position (x0) to a later position (x). Technically, we can displaced say that the car was ______________ from its original position to its final position. Its displacement is Δx, final which is simply defined as the ___________ initial position minus the ____________ position, or Δx = (x − x0). If a car is traveling at a constant velocity v0, this displacement equals the 𝑥 − 𝑥0 = 𝑣0 𝑡 velocity _____________ time times ________: If the car experiences constant acceleration (a), it is shown by a mathematically 1 𝑥 − 𝑥0 = 𝑣0 𝑡 + 𝑎𝑡 2 added derived term ____________ to the equation: 2 final position, x, The general form for the displacement equation solves for the _______ 1 𝑥 = 𝑥0 + 𝑣0 𝑡 + 𝑎𝑡 2 adding by ___________ x0 to both sides: 2 zero In many common problems, one or more of these terms will be equal to ________. 1 𝑥 = 𝑣0 𝑡 + 𝑎𝑡 2 For instance, if the starting position is zero, the first term disappears: 2 zero And if both the initial position and initial velocity are ________, both of the first 1 𝑥 = 𝑎𝑡 2 two terms disappear: 2 Finally, understand that we're using x as a placeholder for _one ______ dimensional 1 motion in this general equation. If the positions, velocities, and accelerations were 𝑦 = 𝑎𝑡 2 2 all in the _y___ direction, the equation would be: Suppose we place a box on a roller conveyor that has an initial velocity of +10 meters/second and a constant acceleration of -1 meter/second2. How far will the box have moved after 10 seconds? The general displacement equation is: 1 𝑥 = 𝑥0 + 𝑣0 𝑡 + 𝑎𝑡 2 2 Since we just want to know total displacement, we can 1 0 𝑥 = 𝑣0 𝑡 + 𝑎𝑡 2 set the initial position, x0, as x0 = _____ and get a 2 simplified equation: Substituting the other known values into the m 10 1 -1 m 𝑥 = (+10 ) (______ s) + (______ ) (10 s)2 displacement equation, we get: s 2 s2 100 m + 1 (−100 m) = _______ = ________ 50 m 2 50 meters. The position, or displacement, of the box from its initial position after 10 seconds is _____ Alternate Velocity Equation Our general one dimensional velocity equation for an object under ________________ acceleration is v = v0 + at. If you know both the acceleration and the __________________, you don't need to know the value of t. Here's the alternate velocity equation, the last of our three general kinematics equations: v2 = v02 + 2ax. We can use this equation to solve for v, v0, _____, or ______. As written, it allows us to solve numerical problems in ______ dimension only, where ______ is the placeholder for that dimension. If the _____________ velocity is zero (v0 = 0), this equation simplifies to v2 = _____ax. An airplane on a runway has an initial speed of +15 meters/second. What is its final velocity when it travels 600 meters on the runway with a constant acceleration of +6 meters/second2? The equation is already set up to solve for v. All we need to do is substitute in the known values: v2 = v02 + 2ax v2 = (_____ m/s)2 + 2(6 m/s2)(600 m) 𝑣 = √(______ m/s)2 + 2(6 m/s2 )(600 m) 𝑣 = √_________ (m/s)2 + 7,200 (m/s)2 = _______ m/s Projectile Motion Now let's solve a problem for an object in ________ fall. Suppose we drop a pebble from the top of a tall building. After falling for three seconds, what will its vertical displacement, ______, be? We have the following values (all in y dimension): v0 = 0.0 m/s (the pebble is dropped from ________) a = –9.8 m/s2 (negative, since it's _________) t = 3.0 s We need to find the __________________ of the stone after three seconds. Substituting these values in the 1 displacement equation for the y dimension, 𝑦 = 𝑣0 𝑡 + 𝑎𝑡 2 , we get: 2 1 𝑦 = (_______ m/s)(3.0 s) + (−9.8 m/s2 )(_______ s)2 2 1 = (−9.8 m/s 2 )(________ s2 ) 2 = ________ m So, after three seconds, the pebble would be ______ meters __________ the top of the building.  All objects in free fall descend under the constant acceleration of _____________. This acceleration only slows or stops if there's a force ______________ gravity. On Earth, the most common force opposing free fall is ______ _____________. Air friction opposes travel through air and eventually _______________________ the force of gravity exactly. This results in _______________ velocity, the final velocity of a falling object. Summary Why do objects in free fall have constant acceleration?