Maximizing range

1
Maximizing range

• A particular projectile lands at the same height
  from which it was launched. Assuming the launch
  speed is constant, what launch angle maximizes
  the range? (The range is the horizontal distance
  between the launch point and the landing point.)
  

1. 30
2. An angle less than 45
3. 45
4. An angle more than 45
5. It depends on the mass of the projectile.

2
Maximizing range

• Whats our equation for range?

3
Maximizing range

• Whats our equation for range?
• , where t is the time of flight.
• If we increase the launch angle, what happens to
  vix? What happens to t?

4
Maximizing range

• Whats our equation for range?
• , where t is the time of flight.
• If we increase the launch angle, what happens to
  vix? What happens to t?
• vix decreases, while t increases.

5
The range equation

• This applies only when the landing height is the
  same as the launch height.
• The range is maximum at a launch angle of 45.
  The equation also tells us that q and 90- q
  produce the same range.

6
Maximizing range

• A particular projectile lands at a level that is
  higher than the level from which it was launched.
  Assuming the launch speed is constant, what
  launch angle maximizes the range? (The range is
  the horizontal distance between the launch point
  and the landing point.)

1. 30
2. An angle less than 45
3. 45
4. An angle more than 45
5. It depends on the mass of the projectile.

7
Analyzing the cart on the ramp

• Lets analyze the situation of the cart on the
  ramp.

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Step 1, draw a diagram

• The cart is like a block on a frictionless ramp.

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Step 2, draw a free-body diagram

• Show the different forces acting on the block.

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Step 2, draw a free-body diagram

• There are two forces, the normal force applied by
  the ramp and the force of gravity.

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Step 3, choose a coordinate system

• What is a good coordinate system in this case?

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Step 3, choose a coordinate system

• Lets align the coordinate system with the
  incline.

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Step 4, break forces into components, parallel to
the coordinate axes

• Which force(s) do we have to split into
  components?

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Step 4, break forces into components, parallel to
the coordinate axes

• We just have to break the force of gravity into
  components.

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Step 4, break mg into components

• Draw a right-angled triangle, with sides parallel
  to the axes, and the force as the hypotenuse.
  Where does ? figure into the triangle?

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Step 4, break mg into components

• Its at the top. Now use sine and cosine.

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Step 4, break mg into components

• Sine goes with the component down the slope,
  cosine with the component into the slope.

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Step 4, break mg into components

• The end result we replaced mg by its
  components.

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The full free-body diagram
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Apply Newtons Second Law

• We apply Newtons Second Law twice, once for the
  x-direction and once for the y-direction.
• x-direction y-direction
• Evaluate the left-hand side of
• each equation by looking at the
• free-body diagram.

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Apply Newtons Second Law

• x-direction y-direction