Lesson 3.3 Projectile Motion Essential Question: How do you solve a problem involving projectile motion?

If you throw a ball down the hall, what is the path the ball travels If you throw a ball down the hall, what is the path the ball travels? What shape does it make? What affects the motion of the ball?

When an object is propelled into the air in a direction other than straight up or down, the velocity, acceleration, and displacement of the object do not all point in the same direction. This makes the vector forms of the equations difficult to solve.

What is the best approach to solving these problems? Resolve the vectors into their components This allows you to apply the simpler one-dimensional forms of the equations for each component You can recombine the components to determine the resultant

When a long jumper approaches his jump, he runs along a straight line, which can be called the x-axis. When he jumps, his velocity has both horizontal and vertical components.

A long jumper’s velocity while sprinting along the runway can be represented by a horizontal vector. Once the jumper is airborne, the jumper’s velocity at any instant can be described by the components of the velocity.

What is projectile motion? The curved path that an object follows when thrown, launched, or otherwise projected near the surface of the Earth

What is a projectile? Any object that is thrown or otherwise projected into the air Some examples of projectiles are softballs, footballs, and arrows

What path does a projectile follow? The path of a projectile is a curve called a parabola and is called its trajectory. With air resistance, a projectile slows down as it collides with air particles

Projectile motion is free fall with an initial horizontal velocity. Gravity will impact the flight of the projectile.

This is a strobe photograph of two table-tennis balls released at the same time. Even though the yellow ball is given an initial horizontal velocity and the red ball is simply dropped, both balls fall at the same rate.

Vertical Motion of a Projectile that Falls from Rest 𝑣 𝑦,𝑓 = 𝑎 𝑦 ∆𝑡 𝑣 𝑦,𝑓 2 =2 𝑎 𝑦 ∆𝑦 ∆𝑦= 1 2 𝑎 𝑦 ∆𝑡 2

Horizontal Motion of a Projectile 𝑣 𝑥 = 𝑣 𝑥,𝑖 =constant ∆𝑥= 𝑣 𝑥 ∆𝑡

How do you find the velocity of a projectile at any point during its flight? Find the vector that has the known components. Use Pythagorean Theorem to find the magnitude of the velocity. Use the tangent function to find the direction of the velocity.

The Royal Gorge Bridge in Colorado rises 321 m above the Arkansas River. Suppose you kick a rock horizontally off the bridge. The magnitude of the rock’s horizontal displacement is 45.0 m. Find the speed at which the rock was kicked.

People in movies often jump from buildings into pools People in movies often jump from buildings into pools. If a person jumps horizontally from the 10th floor (30.0 m) to a pool that is 5.0 m away from the building, with what initial speed must the person jump?

A baseball rolls off a 0. 70 m high desk and strikes the floor 0 A baseball rolls off a 0.70 m high desk and strikes the floor 0.25 m away from the base of the desk. How fast was the ball rolling?

A cat chases a mouse across a 1. 0 m high table A cat chases a mouse across a 1.0 m high table. The mouse steps out of the way, and the cat slides off the table and strikes the floor 2.2 m from the edge of the table. When the cat slid off the table, what was its speed?

A pelican flying along a horizontal path drops a fish from a height of 5.4 m. The fish travels 8.0 m horizontally before it hits the water below. What is the pelican’s speed?

A pelican flying along a horizontal path at 7 A pelican flying along a horizontal path at 7.6 m/s drops a fish from a height of 2.7 m above the water, how far would the fish travel horizontally before hitting the water below?

What about an object launched at an angle? If a projectile is launched at an angle, use components to analyze it. These objects will have an initial velocity component that is both horizontal and vertical.

Its vertical motion will depend upon the vertical component of its initial velocity and the acceleration of gravity. Its horizontal motion will depend upon the horizontal component of its initial velocity.

Projectiles Launched at an Angle 𝑣 𝑥 = 𝑣 𝑥,𝑖 = 𝑣 𝑖 𝑐𝑜𝑠𝜃=constant ∆𝑥= 𝑣 𝑖 𝑐𝑜𝑠𝜃 ∆𝑡 𝑣 𝑦,𝑓 = 𝑣 𝑖 𝑠𝑖𝑛𝜃+ 𝑎 𝑦 ∆𝑡 𝑣 𝑦,𝑓 2 = 𝑣 𝑖 2 𝑠𝑖𝑛𝜃 2 +2 𝑎 𝑦 ∆𝑦 ∆𝑦= 𝑣 𝑖 𝑠𝑖𝑛𝜃 ∆𝑡+ 1 2 𝑎 𝑦 ∆𝑡 2

A zookeeper finds an escaped monkey hanging from a light pole A zookeeper finds an escaped monkey hanging from a light pole. Aiming her tranquilizer gun at the monkey, she kneels 10.0 m from the light pole, which is 5.00 m high. The tip of her gun is 1.00 m above the ground. At the same moment that the monkey drops a banana, the zookeeper shoots. If the dart travels 50.0 m/s, will the dart hit the monkey, the banana or neither one?

A golfer practices driving balls off a cliff and into the water below A golfer practices driving balls off a cliff and into the water below. The edge of the cliff is 15 m above the water. If the golf ball is launched at 51 m/s at an angle of 15°, how far does the ball travel horizontally before hitting the water?

In a scene in an action movie, a stuntman jumps from the top of one building to the top of another building 4.0 m away. After a running start, he leaps at a velocity of 5.0 m/s at angle of 15° with respect to the flat roof. Will he make it to the other roof, which is 2.5 m shorter than the building he jumps from?

A golfer hits a golf ball at an angle of 25. 0° to the ground A golfer hits a golf ball at an angle of 25.0° to the ground. If the golf ball covers a horizontal distance of 301.5 m, what is the ball’s maximum height? (Hint: At the top of its flight, the ball’s vertical velocity component will be zero)

A baseball is thrown at an angle of 25° relative to the ground at a speed of 23.0 m/s. If the ball was caught 42.0 m from the thrower, how long was it in the air? How high did the ball travel before being caught?

Salmon often jump waterfalls to reach their breeding grounds Salmon often jump waterfalls to reach their breeding grounds. One salmon starts 2.00 m from a waterfall that is 0.55 m tall and jumps at an angle of 32.0°. What must be the salmon’s minimum speed to reach the waterfall?

How do you solve problems involving projectile motion?