# Calculating Speed of Point P in Physics Homework

• Saitama
In summary, the author attempted to solve for the velocity of point P in terms of the coordinates at time t, but was unable to do so using basic algebra. He was then able to use the Pythagorean theorem to find the resultant speed.
Saitama

(see attachment)

## The Attempt at a Solution

I was able to calculate the velocity of point P in x direction (to the left). For the speed, I need to find the velocity in y direction (vertically up). Here is my attempt:
At t=0, the distance of point P from O is l, let at time t, the angle POR be ##\theta##. The displacement is ##y=l(\sin \theta -1)##. Differentiating this equation, ##\frac{dy}{dt}=l \cos \theta \cdot \frac{d \theta}{dt}##. The term dθ/dt is equal to the angular velocity but I don't know the radius of curvature of its path which I have to use in the formula ω=vr, where ω is the angular velocity, v is the velocity in x direction and r is the radius of curvature.

Any help is appreciated. Thanks!

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How far does P get from O after t =0?

tms said:
How far does P get from O after t =0?

##l \cos \theta##.

Pranav-Arora said:

(see attachment)

## The Attempt at a Solution

I was able to calculate the velocity of point P in x direction (to the left). For the speed, I need to find the velocity in y direction (vertically up). Here is my attempt:
At t=0, the distance of point P from O is l, let at time t, the angle POR be ##\theta##. The displacement is ##y=l(\sin \theta -1)##. Differentiating this equation, ##\frac{dy}{dt}=l \cos \theta \cdot \frac{d \theta}{dt}##. The term dθ/dt is equal to the angular velocity but I don't know the radius of curvature of its path which I have to use in the formula ω=vr, where ω is the angular velocity, v is the velocity in x direction and r is the radius of curvature.

Any help is appreciated. Thanks!

You're overcomplicating things. Let ∠SUT = 2θ (so you're looking for what happens when θ = 45 deg).

Take O as the origin.

Let ##u_x, u_y, p_x, p_y## be (respectively), the co-ordinates of U and P at time t.

Can you find simple expressions for those in terms of l and θ? (except that ##u_y## is simply constant at zero).

Can you find ##\frac{du_x}{dt}## in terms of ##\frac{d\theta}{dt}## using Chain Rule? Hence rearrange to determine a numerical value for ##\frac{d\theta}{dt}## at the instant of interest.

Can you differentiate ##p_x, p_y## wrt t to find the instantaneous horizontal and vertical velocities of P?

Now use Pythagoras theorem to find the resultant speed.

Find the coordinates of P in terms of the angle. The velocity components are the time derivatives of coordinates.

ehild

Edit:Curious beat me ...

Curious3141 said:
You're overcomplicating things. Let ∠SUT = 2θ (so you're looking for what happens when θ = 45 deg).

Take O as the origin.

Let ##u_x, u_y, p_x, p_y## be (respectively), the co-ordinates of U and P at time t.

Can you find simple expressions for those in terms of l and θ? (except that ##u_y## is simply constant at zero).

Can you find ##\frac{du_x}{dt}## in terms of ##\frac{d\theta}{dt}## using Chain Rule? Hence rearrange to determine a numerical value for ##\frac{d\theta}{dt}## at the instant of interest.

Can you differentiate ##p_x, p_y## wrt t to find the instantaneous horizontal and vertical velocities of P?

Now use Pythagoras theorem to find the resultant speed.

Thanks a lot Curious, that solved the problem.

The velocity of P in both the directions come out to be same!

Pranav-Arora said:
Thanks a lot Curious, that solved the problem.

The velocity of P in both the directions come out to be same!

That's because the sine and cosine of 45 deg are the same!

tms said:
How far does P get from O after t =0?
Pranav-Arora said:
##l \cos \theta##.
Not the x coordinate, but the distance from O. I was just trying to point out that P moves in a circle.

tms said:
Not the x coordinate, but the distance from O. I was just trying to point out that P moves in a circle.

Yep, didn't notice that at all. Thanks!

## 1. What is the formula for calculating speed?

The formula for calculating speed is: Speed = Distance / Time.

## 2. How do I find the distance traveled?

To find the distance traveled, you can use the formula: Distance = Speed x Time.

## 3. What unit of measurement is used for speed?

The most commonly used unit for speed is meters per second (m/s), but other units such as kilometers per hour (km/h) or miles per hour (mph) may also be used.

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

Speed is a scalar quantity that measures the rate of motion without considering direction, while velocity is a vector quantity that measures the rate of motion in a specific direction.

## 5. Can speed be negative?

Yes, speed can be negative if the direction of the motion is opposite to the chosen positive direction. For example, if a car is moving backwards, its speed would be negative.

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