Quadratic EquationsProjectile Motion

As you might know, penguins can't fly – their wings are specially adapted for swimming. However, you can help them getting around my sling-shotting them across the ice:

Given the previous chapters, you might notice that the flight paths taken by the penguins look suspiciously like a parabolasellipsessemicircles. But why?

Let us try to predict the motion of the penguin using the laws physics. The key observation is that we can split the curved motion of the penguin into two completely separate parts:

The vertical speed changes because the force of gravity is pulling the ball back to Earth. The rate of change of speed is often called acceleration, and on Earth it is always −10ms2 (we use a negative value because gravity acts downwards).

| Time | 0 | 1s | 2s | 3s | 4s | 5s | 6s | 7s | 8s | 9s | | Acceleration | -10 | -10 | -10 | -10 | -10 | -10 | -10 | -10 | -10 | -10 | | Speed | 50 | 40 | 30 | 20 | 10 | 0 | -10 | -20 | -30 | -40 | | Position | 0 | 50 | 90 | 120 | 140 | 150 | 150 | 140 | 120 | 90 | {.grid}

The result, as you can see, is a parabola. Using Calculus

https://nssdc.gsfc.nasa.gov/planetary/lunar/apollo_15_feather_drop.html

Of course, flying penguins are not the only things that move on a parabolic path. If we ignore other forces like friction, wind or air resistance, any that flies through the air and is affected by gravity then all objects we throw into the air follow a parabolic path: including basketballs, jets of water, or even rockets.

Here you can see a few more examples:

Here are examples:

Since we know the initial speed of the ball, we can easily calculate the speed after every subsequent second.

Speed, on the other hand, is the rate of change of position. The ball started at a height 0 when we threw it, and we can calculate its height after every subsequent second by adding the speed at that time.

Let’s start with a simple example: a ball is thrown straight up into the air, with an initial speed of 50m/s. Using the laws of physics, we can predict the motion of the ball, and the time it will take for the ball to fall back on the ground.

Galileo used this kind of idea to deduce that because acceleration due to gravity is constant, and acceleration is a kind of double difference, all objects propelled through the air follow the paths of quadratic graphs.

PRACTICE 119: Lizzy throws a ball into the air. Its height, in feet, at time t seconds is given by the quadratic formula:

H(t)=−32t2+480t+6.

a) Find H(0). What does this number mean? b) At what time is the ball at its maximum height? c) When does the ball hit the ground?

Having used quadratic equations to create the optimal business plan for our Skateboard company, and to design the largest possible skate park, now lets actually XXX.

When practicing jumps and tricks, it is important to understand how gravity

skateboarding image