Projectiles

Chapter 5

If we ignore the effects of air resistance an object in the air has a horizontal acceleration of ax = 0 m/s2 and a vertical acceleration of ay = −9.8 m/s2.

Since ax = 0 the horizontal velocity is constant throughout the flight.The vertical acceleration of −9.8 m/s2 causes the vertical velocity to decrease on the way up, be 0 at the peak of the flight and increase in the negative direction on the way down.

𝑉𝑦 𝑓=𝑉𝑦𝑖+𝑎𝑡

(𝑉𝑦 𝑓 )2=(𝑉𝑦 𝑖)

2+2𝑎(𝑦 𝑓 − 𝑦 𝑖)

𝑦 𝑓=𝑦 𝑖+𝑉𝑦𝑖 (𝑡 )+ 12𝑎(𝑡)2𝑥=𝑉𝑥(𝑡)

Explanation of the Variables for Projectile Motion

a = −9.8 m/s2

a is vertical accelerationVyi Initial Vertical VelocityVyf Final Vertical Velocityyi Initial Heightyf Final Heightt Timex Horizontal Distance

If we assume air resistance is negligible, then Ax horizontal acceleration is zero.

𝑉𝑦 𝑓=𝑉𝑦𝑖+𝑎𝑡

(𝑉𝑦 𝑓 )2=(𝑉𝑦 𝑖)

2+2𝑎(𝑦 𝑓 − 𝑦 𝑖)

𝑦 𝑓=𝑦 𝑖+𝑉𝑦𝑖 (𝑡 )+ 12𝑎(𝑡)2

𝑥=𝑉𝑥(𝑡)

Vertical Motion Equations

Horizontal Motion Equation

𝑉𝑦 𝑓=𝑉𝑦𝑖+𝑎𝑡Vertical Motion Equations

• Use to solve for time between velocity points.

• Use to solve for initial or final velocity when given time.

(𝑉𝑦 𝑓 )2=(𝑉𝑦 𝑖)

2+2𝑎(𝑦 𝑓 − 𝑦 𝑖)

Vertical Motion Equations

Use to find any of the following variablesVyi Initial Vertical VelocityVyf Final Vertical Velocityyi Initial Heightyf Final Height

Vertical Motion Equations

This equation is used to find the height at some time point.

𝑥=𝑉𝑥(𝑡)

This equation can be used to find: • The horizontal distance traveled in some time interval.• The time to travel a horizontal distance.• The horizontal velocity needed to travel a distance in a

given time interval.

Horizontal Motion Equation

Takeoff and landing height effects the time to the peak and the time down.

A ball is kicked with an initial vertical velocity (Vyi) of 14 m/s, find the vertical velocity of the ball after 2.4 seconds.

A football is punted with an initial vertical velocity (Vyi) of 18 m/s, find the time to the peak.

A golf ball lands with a final vertical velocity (Vyf) of −23 m/s, find the time down (peak to landing).

A javelin is thrown with an initial height (yi) of 1.3 m, initial vertical velocity (Vyi) of 18 m/s, find the peak height (yf).

A baseball lands with a final height (yf) of 0 m, final vertical velocity (Vyf) of −37 m/s, find the peak height (yi).

A pole vaulter falls from peak height (yi) of 4.1 m to a final height (yf) of 1.1 m, find the final vertical velocity (Vyf) of the pole vaulter.

A golf ball is hit with an initial height (yi) of 1.8 m, initial vertical velocity (Vyi) of 23 m/s, a horizontal velocity (Vx) of 32 m/s find the height (yf) of the ball after 3.6 seconds.

Compute the horizontal distance (x) the ball covers during the 3.6 seconds.