Projectile Motion Notes

Projectile Motion Notes

Projectiles - Objects thrown or launched into the air that are subject to gravity.

Projectiles

• Projectile Motion - Free fall with an initial horizontal velocity.

Types of Projectiles

• Projectiles launched horizontally

• Projectiles launched at an angle

Projectiles Launched from Rest

Components of Projectiles

• Projectiles have two components

• x = horizontal component

• y = vertical component

Equations - Vertical Motion

•  $V_{y,\ f}=a_y\Delta t$  : The final velocity in the y direction is equal to the acceleration in the y direction (gravity) times the change in time.

•  $\left(V_{y,\ f}\right)^2=2a_y\Delta t$  : The final velocity in the y direction squared is equal to double the acceleration in the y direction (gravity) times the change in time.

•  $\Delta y=\frac{1}{2}a_y\left(\Delta t\right)^2$  : Displacement in the y direction is equal to 1/2 the acceleration due to gravity times the change in time squared.

Equations - Horizontal Motion

•  $V_x=V_{x,\ i}$  : Horizontal velocity at any point is equal to the initial horizontal velocity. This means the velocity in the x direction is constant!

•  $\Delta x=V_x\Delta t$  Displacement in the x direction is equal to the velocity in the x direction times the change in time.

Example 1

• Suppose you kick a rock off a 321 m tall bridge. The magnitude of the rock's horizontal displacement is 45 m.

  Find the speed at which the rock was kicked.

Example 2

• Jenny rides her bike off an apartment building in an attempt to land in the pool. The four-story building is 12.5m tall, and she rides at a top speed of 11m/s. 

• How long is she in the air?

• If the pool is 5m from the building, and is 10m wide, will she splash down, land short, or fly right over the pool?

Example 3

• Stephanie goes cliff diving. The cliff is 52 meters high. How long would it take for her to hit the water?

• There is a 6.5 meter wide sandbar directly below the cliff. What must her horizontal speed be to clear the sandbar and hit the deep water?

Projectiles Launched at an Angle

Use vector components to analyze projectiles launched at an angle.

Vertical Component

•  $V_{y,\ f}=V_i\ \sin\theta\ +a_y\Delta t$  : Vertical final velocity is equal to the initial velocity times the sine of the launched angle plus the vertical acceleration times the change in time.

•  $\left(V_{y,\ f}\right)^2=V_i^2\ \left(\sin\theta\right)^2+2a_y\Delta y$  : Final vertical velocity squared = initial velocity squared times the sine of the launch angle squared + 2 times the vertical acceleration times the vertical displacement.

•  $\Delta y=\left(V_i\ \sin\theta\right)\Delta t+\frac{1}{2}a_y\left(\Delta t\right)^2$  : Vertical displacement = initial velocity times the sine of the launch angle times the change in time + 1/2 vertical acceleration times the change in time squared.

Horizontal Component

•  $V_x=V_{x,\ i}=V_i\ \cos\theta\$  : The velocity in the horizontal direction is constant. 

• The x-component for velocity can be found by multiplying the initial velocity by the cosine of the launch angle.

Example 1

A baseball is thrown at an angle of 25° relative to the ground at a speed of 23 m/s. If the ball was caught 42 m from the thrower, how long was it in the air? How high did the ball travel before being caught?

Example 2

A cannon ball is fired with an initial velocity of 100.0 m/s at an angle of 45° above the horizontal. What maximum height will it reach and how far will it fly horizontally?

Example 3

Suppose a cannon ball is fired downward from a 50.0 m high cliff at an angle of 45° with an initial velocity of 80.0 m/s. How far horizontally will it land from the base of the cliff?