Calculating Speed of Point P in Physics Homework

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.
  • #1
Saitama
4,243
93

Homework Statement


(see attachment)


Homework Equations





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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  • #2
How far does P get from O after t =0?
 
  • #3
tms said:
How far does P get from O after t =0?

##l \cos \theta##.
 
  • #4
Pranav-Arora said:

Homework Statement


(see attachment)


Homework Equations





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.
 
  • #5
Find the coordinates of P in terms of the angle. The velocity components are the time derivatives of coordinates.

ehild

Edit:Curious beat me ...
 
  • #6
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. :smile:

The velocity of P in both the directions come out to be same!
 
  • #7
Pranav-Arora said:
Thanks a lot Curious, that solved the problem. :smile:

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! :wink:
 
  • #8
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.
 
  • #9
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!
 

Related to Calculating Speed of Point P in Physics Homework

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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