Projectile Motion velocity and direction

[aaronfue]

Homework Statement

At t = 0, a projectile is located at the origin and has a velocity of 20 m/s at 40° above the horizontal. The profile of the ground surface it strikes can be approximated by the equation y = 0.4x – 0.006x², where x and y are in meters. Determine (a) the approximate coordinates of the point where it hits the ground, (b) its velocity and direction when it hits the ground, (c) its highest point in flight and (d) the greatest distance above the ground.

Homework Equations

v_o= 20 $\frac{m}{s}$
θ=40°
Horizontal Motion: x=x_o² + (v_o)_x²t + $\frac{1}{2}$at
Vertical Motion: y=y_o² + (v_o)_y²t + $\frac{1}{2}$at

The Attempt at a Solution

This function of the ground is confusing me. I'm not sure where to start with this.

 

[tms]

When you do one of these problems with a flat ground you also have do deal with a function that describes the height of the ground at a given distance, but it is easy since the function is constant. Here you do the same thing, but you have a non-constant function for the ground. In both cases you find the place where the trajectory intersects the ground.

 

[haruspex]

Horizontal Motion: x=x_o² + (v_o)_x²t + $\frac{1}{2}$at
Vertical Motion: y=y_o² + (v_o)_y²t + $\frac{1}{2}$at

The powers of 2 are in the wrong places.

 

[aaronfue]

The powers of 2 are in the wrong places.

Got it. Thanks!

 

[Nickie]

I would approach this problem by breaking it down into smaller parts and using known equations and principles to solve for the unknowns. First, I would use the initial velocity and angle to calculate the horizontal and vertical components of the projectile's velocity. This can be done using trigonometric functions, such as cos and sin.

Next, I would use the equations for horizontal and vertical motion to solve for the time it takes for the projectile to hit the ground. This can be done by setting the vertical position equal to 0 and solving for time.

Once the time is known, I would use the equations for horizontal and vertical motion again to solve for the coordinates of the point where the projectile hits the ground. This can be done by plugging in the calculated time into the equations and solving for x and y.

To determine the velocity and direction at impact, I would use the calculated time to find the velocity at that point in time using the equations for horizontal and vertical motion. The magnitude of the velocity can be calculated using the Pythagorean theorem and the direction can be found using trigonometric functions.

To find the highest point in flight, I would use the equation for vertical motion and set the velocity in the y-direction equal to 0. This would give the maximum height of the projectile.

Finally, to find the greatest distance above the ground, I would use the equation for horizontal motion and plug in the time at which the projectile hits the ground. This would give the horizontal distance traveled by the projectile before hitting the ground.

Overall, the key to solving this problem would be to use the known equations for projectile motion and to carefully consider the initial conditions and the shape of the ground surface.