Projectile Motion SPH4U
Projectiles A projectile is an object on which the only force acting is gravity. It is said to be “in free fall” even if part of its motion is upward.
Projectiles A projectile may also be travelling horizontally while falling if its initial (launch) velocity had a horizontal component.
Parametric Equations As the vertical motion is accelerated and any horizontal motion is not accelerated, the components are calculated separately:
Parametric Equations As the vertical motion is accelerated and any horizontal motion is not accelerated, the components are calculated separately: Vertical acceleration is –g. Horizontal acceleration is zero.
Parametric Equations As the vertical motion is accelerated and any horizontal motion is not accelerated, the components are calculated separately:
Parametric Equations As the vertical motion is accelerated and any horizontal motion is not accelerated, the components are calculated separately: Note that the variable common to both sets of equations is Dt.
The Parabola Note that the path (or “trajectory”) of a projectile will therefore be parabolic.
Example 1: The Cliff A ball is launched at 15.0 m/s [right] from the top of a 15.0 m tower. How far does the ball travel horizontally before it lands?
Example 1: The Cliff A ball is launched at 15.0 m/s [right] from the top of a 15.0 m tower. How far does the ball travel horizontally before it lands? First, find the time it takes the ball to fall 15.0 m:
Example 1: The Cliff First, find the time it takes the ball to fall 15.0 m:
Example 1: The Cliff First, find the time it takes the ball to fall 15.0 m:
Example 1: The Cliff First, find the time it takes the ball to fall 15.0 m:
Example 1: The Cliff First, find the time it takes the ball to fall 15.0 m:
Example 1: The Cliff Now find the horizontal distance travelled in 1.75 s:
Example 1: The Cliff Now find the horizontal distance travelled in 1.75 s:
The Initial Velocity If the initial velocity is not parallel or perpendicular to the horizontal, it must be broken down into its horizontal and vertical components:
The Range A ball is launched from ground level with an initial velocity of v1 at an angle q with the horizontal. How far does the ball travel horizontally before it lands?
The Range A ball is launched from ground level with an initial velocity of v1 at an angle q with the horizontal. How far does the ball travel horizontally before it lands? Note that the answer to this question will not be a number but an algebraic expression.
The Range
The Range We can discard this solution.
The Range Substituting into the equation for horizontal distance:
The Range Substituting into the equation for horizontal distance:
The Range Substituting into the equation for horizontal distance: (by trig identities)
The Range The advantage of deriving a general expression instead of calculating a number is that you can see the relationships between the variables. Ex. So a projectile on the moon, with 1/6 the gravity, will travel 6 times further.
The Range The advantage of deriving a general expression instead of calculating a number is that you can see the relationships between the variables. Ex. So a projectile on the moon, with 1/6 the gravity, will travel 6 times further. If the launch velocity is doubled, the projectile will travel 4 times further.
The Range You can also find maxima and minima: Which launch angle will give you the maximum horizontal distance?
The Range You can also find maxima and minima: Which launch angle will give you the maximum horizontal distance? The maximum value of sin2q is 1.
The Range You can also find maxima and minima: Which launch angle will give you the maximum horizontal distance? The maximum value of sin2q is 1.
The Range However, the expression only applies to objects launched from ground level. It can not be applied in cases such as:
Example 2: Up Off the Cliff A ball is launched at 15.0 m/s [37o above the horizontal] from the top of a 15.0 m tower. How far does the ball travel horizontally before it lands?
Example 2: Up Off the Cliff A ball is launched at 15.0 m/s [37o above the horizontal] from the top of a 15.0 m tower. How far does the ball travel horizontally before it lands?
Example 2: Up Off the Cliff
Example 2: Up Off the Cliff
Example 2: Up Off the Cliff
Example 2: Up Off the Cliff
Example 2: Up Off the Cliff Note that the ball stayed in the air longer and travelled further when launched slightly above the horizontal.
Example 2: At the Bottom What was the ball’s impact velocity?
Example 2: At the Bottom What was the ball’s impact velocity? Note that it is not zero upon impact. It is only zero after impact, after the ground has exerted a normal force on it to stop it.
Example 2: At the Bottom What was the ball’s impact velocity? Note that we still need to find the time first.
Example 2: At the Bottom What was the ball’s impact velocity? 11.98m/s
Example 2: At the Bottom What was the ball’s impact velocity? 11.98m/s
Example 2: At the Bottom What was the ball’s impact velocity? 11.98m/s
More Practice Textbook Questions p. 46 #3, 5 p. 50 #9, 10 Tomorrow: “Projectile Motion Lab”