# 2007 F=ma Contest problem 17

In summary, the problem involves a point-like object being thrown horizontally off a 50 meter high building with an initial speed of 10 m/s. The object experiences an acceleration component tangential to its trajectory and an acceleration component perpendicular to its trajectory. The question asks for the time at which the tangential acceleration is twice the perpendicular acceleration. After considering different approaches, it is found that the problem can be solved using kinematics, by finding the angle of the velocity vector and using it to calculate the time at which the point will be headed in the desired direction.

## Homework Statement

A small point-like object is thrown horizontally off of a 50 meter high building with an initial speed of 10 m/s. At any point along the trajectory there is an acceleration component tangential to the trajectory and an acceleration component perpendicular to the trajectory. How many seconds after the object is thrown is the tangential component of the acceleration of the object equal to twice the perpendicular component of the acceleration of the object?
Ignore air resistance.

## Homework Equations

Kinematics? I can't seem to be able to mathematically express an "acceleration component perpendicular to the trajectory"

## The Attempt at a Solution

I know that the only force acting on the object is gravity so the acceleration vector would be downwards. However they're talking components, so I presume that this tangential component has an x-component that cancels out with the x-component of the perpendicular component. That way they yield a downward pointing vector...This problem shouldn't be too difficult, but I just don't understand how to approach it...btw the answer is 2.00 seconds...

I tried another approach (probably somewhat flawed) , which didn't delve into the mathematical aspect of the problem, but rather I just drew a diagram of the object's trajectory... initially the acceleration is perpendicular to the starting point of the trajectory and when the object strikes the ground i would think the acceleration would be nearly tangential to the trajectory. however, this makes it sound like the trajectory is a circle (like the quarter of the circle that lies, in say the first quadrant)...so as you can see the overall acceleration vector, i hypothesized, would be more--- say 2 parts tangential to one part perpendicular...wow its hard to convey what i want to say...anyways, i feel like i am overthinking this problem and there should be some formulated, mechanical approach to get the answer.

Any input/comment(s) are appreciated. Thank you for your help.

## Homework Statement

A small point-like object is thrown horizontally off of a 50 meter high building with an initial speed of 10 m/s. At any point along the trajectory there is an acceleration component tangential to the trajectory and an acceleration component perpendicular to the trajectory. How many seconds after the object is thrown is the tangential component of the acceleration of the object equal to twice the perpendicular component of the acceleration of the object?
Ignore air resistance.

## Homework Equations

Kinematics? I can't seem to be able to mathematically express an "acceleration component perpendicular to the trajectory"

## The Attempt at a Solution

I know that the only force acting on the object is gravity so the acceleration vector would be downwards. However they're talking components, so I presume that this tangential component has an x-component that cancels out with the x-component of the perpendicular component. That way they yield a downward pointing vector...This problem shouldn't be too difficult, but I just don't understand how to approach it...btw the answer is 2.00 seconds...

I tried another approach (probably somewhat flawed) , which didn't delve into the mathematical aspect of the problem, but rather I just drew a diagram of the object's trajectory... initially the acceleration is perpendicular to the starting point of the trajectory and when the object strikes the ground i would think the acceleration would be nearly tangential to the trajectory. however, this makes it sound like the trajectory is a circle (like the quarter of the circle that lies, in say the first quadrant)...so as you can see the overall acceleration vector, i hypothesized, would be more--- say 2 parts tangential to one part perpendicular...wow its hard to convey what i want to say...anyways, i feel like i am overthinking this problem and there should be some formulated, mechanical approach to get the answer.

Any input/comment(s) are appreciated. Thank you for your help.

ok now that I read it further its not quite as I posted.
hmmmm...

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## Homework Statement

A small point-like object is thrown horizontally off of a 50 meter high building with an initial speed of 10 m/s. At any point along the trajectory there is an acceleration component tangential to the trajectory and an acceleration component perpendicular to the trajectory. How many seconds after the object is thrown is the tangential component of the acceleration of the object equal to twice the perpendicular component of the acceleration of the object?
Ignore air resistance.

## Homework Equations

Kinematics? I can't seem to be able to mathematically express an "acceleration component perpendicular to the trajectory"

## The Attempt at a Solution

I know that the only force acting on the object is gravity so the acceleration vector would be downwards. However they're talking components, so I presume that this tangential component has an x-component that cancels out with the x-component of the perpendicular component. That way they yield a downward pointing vector...This problem shouldn't be too difficult, but I just don't understand how to approach it...btw the answer is 2.00 seconds...

I tried another approach (probably somewhat flawed) , which didn't delve into the mathematical aspect of the problem, but rather I just drew a diagram of the object's trajectory... initially the acceleration is perpendicular to the starting point of the trajectory and when the object strikes the ground i would think the acceleration would be nearly tangential to the trajectory. however, this makes it sound like the trajectory is a circle (like the quarter of the circle that lies, in say the first quadrant)...so as you can see the overall acceleration vector, i hypothesized, would be more--- say 2 parts tangential to one part perpendicular...wow its hard to convey what i want to say...anyways, i feel like i am overthinking this problem and there should be some formulated, mechanical approach to get the answer.

Any input/comment(s) are appreciated. Thank you for your help.

Assuming gravity is the only force acting on the point you could split up the acceleration due to g into two components relative to the current "path". The angle of the velocity vector that would give you this ratio could be found. And then you could find the time at which the point would be headed in this direction/angle. looks like the velocity vector would be headed at a 27.5 degree or 63.5 degree angle with respect to the real vertical or horizontal. If I am reading this correctly.

## 1. What is the 2007 F=ma Contest problem 17?

The 2007 F=ma Contest problem 17 is a physics problem that was given as part of the annual Physics Olympiad competition for high school students in the United States. It is a multiple-choice question that tests students' understanding of Newton's laws of motion and their ability to apply them to a real-world scenario.

## 2. What is the premise of problem 17?

The problem presents a scenario where a car is driving on a circular track with a constant speed. The car then suddenly hits a patch of ice and begins to slide outward. The question asks students to calculate the acceleration of the car at this point.

## 3. What are the main concepts being tested in this problem?

This problem primarily tests students' understanding of centripetal force and Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

## 4. How can one approach solving this problem?

To solve this problem, one would need to first draw a free-body diagram of the car at the point where it hits the patch of ice. This would help identify all the forces acting on the car, including the normal force, friction force, and centripetal force. Then, using Newton's second law, one can set up an equation and solve for the acceleration.

## 5. What are some tips for students attempting this problem?

Some tips for solving this problem include drawing a clear and accurate free-body diagram, carefully considering all the forces at play, and being mindful of units when setting up and solving the equation. It can also be helpful to practice similar problems beforehand to familiarize oneself with the concepts and calculations involved.