# Kinematics: How fast was he when he got off the ramp

• ramb
In summary, the conversation discussed finding the speed of a bike jumping between two ramps, given the height, angles, and distance between the ramps. The participants discussed using the equation 1/2 * at^2 + vi*t + di = df and separating it into x and y components, leading to a quadratic equation in t. They also discussed the substitution of t and the use of a trigonometric identity to simplify the equation and solve for the initial velocity.
ramb

## Homework Statement

A bike is jumping between two ramps. The ramps are a height H , both with angles $$\theta$$ and separated by a distance D. If he landed halfway down the landing ramp find the speed at which he left the launching ramp in terms of H, $$\theta$$ and D.

## Homework Equations

1/2 * at^2 + vi*t + di = df
(i'm sure you all know that one)

## The Attempt at a Solution

Okay so here's what I did, but apparently it was completely off, but i don't know where to begin it. *see attachment*

#### Attachments

• IMG00191.jpg
19.5 KB · Views: 359
I like that eqn.

We know the takeoff angle. D is straightforward, how do we express the landing point in terms of the variables given. We are told it is 1/2 way down the ramp, so what is df and di. What are the initial x and y velocities?

How can we figure the time of flight?

The landing point is:

$$\frac{H}{tan(\theta}$$

Let
$$L$$ = $$\frac{H}{tan(\theta}$$

With the equation
$$\frac{1}{2}at^{2}+v_{i}t+d_{i}$$

I separated all of it to x and y components and got:

$$v_{i}t=-\frac{at^{2}}{2}+d_{f}$$
then

$$v_{i}=-\frac{at^{2}}{2t}+\frac{d_{f}}{t}$$

so

$$v_{i}_{x}=-\frac{at}{2}+\frac{D+\frac{L}{2}}{t}$$

and

$$v_{i}_{y}=-\frac{gt}{2}-\frac{H}{2t}$$

Once I have both of them, do i just have to square both of them, put them under the square route then, finish?

$$v_{i}=\sqrt{(-\frac{at}{2}+\frac{D+\frac{L}{2}}{t})^{2}+(-\frac{gt}{2}-\frac{H}{2t})^{2}}$$

then substituting for L
$$v_{i}=\sqrt{(-\frac{at}{2}+\frac{D+\frac{(\frac{H}{tan(\theta)})}{2}}{t})^{2}+(-\frac{gt}{2}-\frac{H}{2t})^{2}}$$

Last edited:
I'll need a sec to look at this, just got back. Stay tuned.

here is the approach I have taken, so far unfruitful because of the 1/2 ramp displacement which bungs up what is usually very clean algebra: let a = angle of the ramp.

Delta x= V Cos(a)*t=.5H/tan(a) + d

Delta y = 1/2H=1/2 gt^2+ V sin(a)*t (a quadratic in t but only the longer time will work)

Normally what I do here would be substitute for t which leads to:

let d+0.5H/tan(a)= C; then 1/2H=1/2 g (C/(V*cos(a)))^2 + (V sin(a)*C/V cos(a))
" = " + C (tan(a))
There may be a trig ID here which can be uses to simplify things considerably.

Thanks,

so why did you substitute the t, and where did you get the C from?

oh I see,

so with the time it took in the x direction, you used that time in terms of v,d,h,theta to put it in the y equation, since that's the one where the time in air matters. So after all that, the only thing is solve for V?

## 1. What is kinematics?

Kinematics is the branch of mechanics that studies the motion of objects without considering the forces that cause the motion.

## 2. Why is kinematics important?

Kinematics allows us to describe and analyze the motion of objects in a mathematical and precise way, which is crucial in many fields such as engineering, physics, and biomechanics.

## 3. What is the difference between speed and velocity?

Speed is a measure of how fast an object is moving, while velocity is a measure of how fast an object is moving in a specific direction. Velocity takes into account both the speed and direction of an object's motion.

## 4. How is acceleration related to kinematics?

Acceleration is the rate of change of an object's velocity over time. In kinematics, acceleration is used to describe how an object's velocity changes as it moves.

## 5. How is kinematics used in real-life applications?

Kinematics is used in many real-life applications, such as designing vehicles and machines, predicting the trajectory of projectiles, and understanding the movement of the human body in sports and rehabilitation.

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