Projectile motion from an angled slope

In summary, the question is asking for the angle at which a rock should be thrown from the top of a slope to achieve maximum range. Using the equations of motion and setting up a new coordinate system, the range can be expressed as a function of the launch angle. The goal is to find the maximum value of this function by solving for the angle with respect to the x-axis.
  • #1
BarneyStinson
10
0

Homework Statement



A rock is thrown from the top of a slope that makes an angle of 30 degrees with the horizontal. At what angle to the horizontal should the rock be thrown to get a maximum range? (Hint: pick the direction of the slope as a new x-axis and the normal to the slope as the new y axis. There will be acceleration in both axes.)

Homework Equations



y=y_o+v_y_o*-.5*g*t^2
v_y=v_y_o+a_y*t

x=x_o+v_x_o+.5a*t^2
v_x=v_x_o

The Attempt at a Solution



I drew a diagram of what is happening:

2wddod1.jpg


I need to solve for theta.

I don't think it will be 45 degrees to the horizontal, as you start on a slant. Even if i did think it was, i don't know where to plug 45 into.

Can anyone guide me on the right path? Offer any more hints?

Thanks.
 
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  • #2
You'll have to write the equations of motion for the projectile in the given coordinate system and find an expression for the range (the x coordinate when the y coordinate reaches zero --- the projectile lands). The trick will be in maximizing this expression with respect to the launch angle. I think it would be easier to first solve for the angle with respect to the x-axis. It can always be adjusted to the horizontal reference afterwards.
 

1. What is projectile motion from an angled slope?

Projectile motion from an angled slope is the motion of an object that is launched at an angle from a horizontal surface. The object follows a curved path due to the combination of its horizontal and vertical velocities.

2. How is the angle of the slope related to the trajectory of the projectile?

The angle of the slope is directly related to the angle of the projectile's initial velocity. The steeper the slope, the greater the angle of the initial velocity and the higher the projectile will travel.

3. What factors affect the range of a projectile launched from an angled slope?

The range of a projectile launched from an angled slope is affected by the initial velocity, angle of launch, and the gravitational force acting on the object. Air resistance and wind can also have an impact on the range.

4. Can the initial velocity and angle of launch be adjusted to achieve the same range on different angled slopes?

Yes, the initial velocity and angle of launch can be adjusted to achieve the same range on different angled slopes. This can be done by using mathematical equations and adjusting the values of the initial velocity and angle to find the desired range.

5. How is the motion of a projectile from an angled slope different from that of a horizontal surface?

The motion of a projectile from an angled slope is different from that of a horizontal surface because the object experiences both horizontal and vertical motion. On a horizontal surface, the object only experiences horizontal motion. Additionally, the angle of the initial velocity and the range of the projectile will be different on an angled slope compared to a horizontal surface.

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