Kinematic's Equations

Kinematic equations can be used to solve many motion problems. Two kinematic equations we are going to look at in this article are 2ad=V_f²-V_i² and V_f=V_i+at: where a is acceleration, d is distance, V_f is final velocity, V_i is initial velocity, and t is time according to Equations of motion from Wikipedia the free encyclopedia. We are going to use these equations to solve for the distance a fired bullet will travel upward, and the amount of time it takes to reach its maximum height. We will solve these problems neglecting the effects of air resistance.

If a bullet is fired from a gun straight up into the air at a velocity of 240.0 meters/second, then how high will it go? The first thing to do when solving this type of problem is to identify the knowns. Well, we know that the initial velocity (V_i) is 240.0 meters/second. We also know that at the top of the path, the bullet will momentarily stop. So, the final velocity (V_f) is 0 meters/second. We also know that any object moving upward accelerates at -9.80665 meters/second² (a) according to Earth's Gravity from Wikipedia the free encyclopdia. The acceleration is negative, because the object is decelerating. An equation that relates the knowns to the unknown (distance) is as follows: 2ad=V_f²-V_i². Plugging in the known values gives the following: 2 (-9.80665) (d) =0²-(240.0)². So, -19.61d= -(240.0)². Simplifying gives the following: -19.61d=-57600. Dividing each side by 19.61 gives the following: d=2937. So, the bullet travels a distance of 2937 meters.

Well, how long does the bullet take to reach maximum height? Well, we know the initial velocity, final velocity, acceleration, and distance. So, an equation that relates the knowns to the unknown (time) is as follows: V_f=V_i+at. Plugging in the knowns gives the following: 0=240.0+ (-9.80665)t. Subtracting 240.0 from each side of the equation gives the following: -240.0=-9.80665t. Dividing each side by -9.80665 gives the following: 24.47 seconds. So, it takes 24.47 seconds to reach maximum height. The bullet is in the air 2 x 24.47 seconds=48.94 seconds. Remember the bullet must spend an equal amount of time going up as it goes down.

In the absence of air resistance what would the maximum speed back down be? Use the equation V_f=V_i+at. This time the initial velocity is 0, because at the top of the path it's momentarily stopped. The acceleration is 9.80665 meters/second². We know the time is 24.47 seconds. Remember it spends and equal amount of time going up as it does going down. So, we are solving for V_f. Plugging all the knowns into the equation gives the following: V_f=0+9.80665(24.47). V_f=240.0 meters/second. This is the same value as the velocity the bullet was fired at!

Kinematic equations can be used to solve many motion problems. Two kinematic equations we looked at in this article were 2ad=V_f²-V_i² and V_f=V_i+at.

Equations of motion from Wikipedia the free encyclopedia, Wikipedia
Earth's Gravity from Wikipedia the free encyclopdia, Wikipedia