Kinematics (conceptual questions)

[vu10758]

I am not sure if this is the right place to ask these questions, but please help me if you can.

1) A projectile is fired an angle x at a speed v_o. The point it is fired from also happens to be the start of an incline which is at an angle y (x>y). Determine the distance up the incline where the projectile lands.

How do I approach this problem? I have no idea how to start.

2) A baseball is thrown straight up into the air. It passes point A with the speed v and point B at a distance d higher with the speed of v/2. How much higher will the ball rise before falling? ( Hint: you don't need to do any calculations)

I know that v = v_o -9.8t. Therefore, v is decreasing at a constant rate. If it takes t seconds to decrease by 1/2, then it will also take the same t second to reach zero. However, how do I express this as a distance.

3) A rock is dropped off of a high cliff. The sound of it striking the ocean is heard a time T later. Assume the speed of sound is V_sound. Determine the height of the cliff.

I know that T = t_1 + t_2 where t_1 is the time it takes the rock to hit the ocean while t_2 is the time sound travels.

The fall is 0 = H + 1/2 v_o * t - 1/2 gt^2
t= SQRT(2H/g)

H = V_sound (T - SQRT (2H/g))

How do I find H from here?

4) A projectile is fired with speed V_o at angle x. Find the maximum value of x such that the projectile's distance from the origin is always increasing up until the time when it hits the ground.

I know that the maximum range is at 45 degree, but how do I find this? The maximum value includes both the horizontal and vertical values of displacement.

I know I asking a lot of questions, but I really need to know these concepts for a test coming up. Please get me started on some of these problems. Any help will be greatly appreciated. Thanks in advance.

 

[Astronuc]

1) A projectile is fired an angle x at a speed v_o. The point it is fired from also happens to be the start of an incline which is at an angle y (x>y). Determine the distance up the incline where the projectile lands.

One develops two equations, one for the projectile trajectory, altitude in terms of distance, and the other equation for the incline, which is also altitude (height) as a function of distance. Then determine where they intersect.

2. A baseball is thrown straight up into the air. It passes point A with the speed v and point B at a distance d higher with the speed of v/2. How much higher will the ball rise before falling? ( Hint: you don't need to do any calculations)

How about conservation of energy (assuming no or negible air resistance). The change in kinetic energy = change in gravitation potential energy. For example, an object starts with vertical velocity V at elevation H1 and rises to H2 where its vertical velocity is 0. Then KE1 + PE1 = KE2 + PE2, or KE1-KE2 = PE2 - PE1, and then substituting the values, mV²/2 - 0 = mgH2-mgH1, and with m dividing out, V²/2 = g (H2-H1).

3) A rock is dropped off of a high cliff. The sound of it striking the ocean is heard a time T later. Assume the speed of sound is V_sound. Determine the height of the cliff.

With zero initial velocity, the rock just falls so H=g(t₁)², which gives t₁ = SQRT(2H/g).

Then T - t₁ = t₂,

and H = V_sound (T - SQRT (2H/g)) as one wrote.

As written, this is a transcendental equation, and there are various ways to solve it. Ideally, one finds an analytical solution. The way to do that is to rearrange the terms and to write it as a quadratic equation in terms of H.

4) A projectile is fired with speed V_o at angle x. Find the maximum value of x such that the projectile's distance from the origin is always increasing up until the time when it hits the ground.

This is the maximum range problem, in which one find the angle for maximum range. If angle is 'a', and range is R, then one develops R(a) and then uses dR(a)/da = 0.

 

[kyle8921]

1) For this problem, you can use the equations of motion for projectile motion to solve for the distance up the incline. Remember that the projectile has both horizontal and vertical components of motion, so you will need to break down the problem into two parts - the horizontal and vertical motion. Use the equations v = u + at and s = ut + 1/2at^2 to solve for the horizontal and vertical displacements, and then use trigonometry to find the distance up the incline.

2) You are on the right track with using the equation v = v_o - 9.8t. To find the distance the ball rises, you can use the equation s = ut + 1/2at^2, but instead of solving for t, you can solve for the maximum height reached by setting the final velocity to 0. This will give you the equation s = v^2/2g. You can then use this equation to find the difference between the heights at points A and B.

3) This problem involves both the motion of the rock and the sound. You have correctly identified the equation T = t_1 + t_2, but you will also need to use the equations of motion for the rock and the speed of sound to solve for the height of the cliff. Set the equations for the rock's motion and the sound's motion equal to each other and solve for H.

4) To find the maximum value of x, you can use the equation for the range of a projectile, which is R = (v_o^2/g)sin(2x). To find the maximum value, set the derivative of this equation with respect to x equal to 0 and solve for x. This will give you the angle at which the range is maximized.

I hope this helps get you started on these problems. Remember to always break down the problems into smaller parts and use the appropriate equations for each aspect of the motion. Good luck on your test!