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.