# Guy pushing girl on a swing models h.displacement with function

• aeromat
In summary, Lee pushes his little sister Kara to swing at a certain height for a certain duration in order to affect her horizontal displacement as a function of time. After a certain number of swings, her displacement diminishes to 1 meter.
aeromat
1,2,3. Homework Statement , and work done.
After Lee gives his little sister Kara a big push on a swing, her horizontal position as a function of time is given by the equation $$x(t) = 3cost(t)*e^{-0.05t}$$ , where x(t) is her horizontal displacement, in metres, from the lowest point of her swing, as a function of time, t, in seconds.

a) From what horizontal distance from the bottom of Kara's swing did Lee push his sister?
I said 3m.

b) Determine the highest speed Kara will reach and when this occurs.
$$x'(t) = -3sin(t)*e^{-0.05t} + 3cos(t)*0.05*e^{-0.05t}$$
$$x'(t) = e^{0.05t}(-3sin(t) - 0.15cos(t))$$
$$x'(t) = -e^{0.05t}(3sin(t) + 0.15cos(t))$$

So,
$$0 = -e^{0.05t}$$ or $$0 = 3sin(t) + 0.15cos(t)$$
$$0 = -e^{0.05t}$$ DNE
$$0 = 3sin(t) + 0.15cos(t)$$
$$-3sin(t) = 0.15cos(t)$$
$$-3tan(t) = 0.15$$
$$tan(t) = 0.15/-3$$
$$t = arctan(0.15/-3)$$

But this gives t = -2.8624; a negative time value.

What did I do wrong?

Part C) How long did it take for Kara's maximum horizontal displacement at the top of her swing arc to diminish to 1m? After how many swings will this occur?

Part D) Sketch the graph <--- I am unsure as to what scale I should use.

you seem to have missed some minus signs in your initial derivative, first in the multiplier after differentiating the exponential (though i think you put that in next step) and then in the exponential itself

now following you working through
$$x(t) = 3 cos(t) e^{-\frac{t}{20}}$$
differentiating
$$x'(t) = -3sin(t) e^{-\frac{t}{20}} + \frac{-1}{20}3cos(t) e^{-\frac{t}{20}}$$
$$x'(t) = -3e^{-\frac{t}{20}}(sin(t) + \frac{1}{20}cos(t))$$

then
$$x'(t) = 0 \ \ \to \ \ sin(t) + \frac{1}{20}cos(t) = 0$$

giving
$$tan(t) = \frac{sin(t)}{cos(t)} = -\frac{1}{20}$$

which is what you have, however setting the first derivative equal to zero will maximise the horizontal distance, not the speed

Last edited:
http://www.wolframalpha.com/input/?i=tan(t)
also note if you drew a line y= -1/20 it would intesect tan(x) infinite times, the first positive value is close to pi, eahc one of these corresponds to local maxima in position

aeromat said:
1,2,3. Homework Statement , and work done.
After Lee gives his little sister Kara a big push on a swing, her horizontal position as a function of time is given by the equation $$x(t) = 3cost(t)*e^{-0.05t}$$ , where x(t) is her horizontal displacement, in metres, from the lowest point of her swing, as a function of time, t, in seconds.

a) From what horizontal distance from the bottom of Kara's swing did Lee push his sister?
I said 3m.

b) Determine the highest speed Kara will reach and when this occurs.
$$x'(t) = -3sin(t)*e^{-0.05t} + 3cos(t)*0.05*e^{-0.05t}$$
$$x'(t) = e^{0.05t}(-3sin(t) - 0.15cos(t))$$
$$x'(t) = -e^{0.05t}(3sin(t) + 0.15cos(t))$$

So,
$$0 = -e^{0.05t}$$ or $$0 = 3sin(t) + 0.15cos(t)$$
$$0 = -e^{0.05t}$$ DNE
$$0 = 3sin(t) + 0.15cos(t)$$
$$-3sin(t) = 0.15cos(t)$$
$$-3tan(t) = 0.15$$
$$tan(t) = 0.15/-3$$
$$t = arctan(0.15/-3)$$

But this gives t = -2.8624; a negative time value.

What did I do wrong?

Part C) How long did it take for Kara's maximum horizontal displacement at the top of her swing arc to diminish to 1m? After how many swings will this occur?

Part D) Sketch the graph <--- I am unsure as to what scale I should use.

You want to find where x'(t) is largest, so you need to look at its *derivative*. When you set its derivative (d/dt) x'(t) to zero, you are finding either a local minimum or local maximum---both have derivative = 0. You need other tests to determine which points are minima and which are maxima; there will be infinitely many points of either type (that is, *local* optima), but among all the local maxima, one will stand out at the absolute maximum. This will all be clear if you plot x'(t) for a large enough range of t.

In your work above you found times where the horizontal speed = 0, and these occur at times t where tan(t) = -1/20. There are infinitely many such t, all differing by multiples of pi (because tan(w+n*pi) = tan(w) for n = 1,2,3,4,... ).

RGV

## 1. What is the concept behind the "Guy pushing girl on a swing models h.displacement with function"?

The concept behind this model is to use mathematical functions to represent the displacement of a girl on a swing as a result of a boy pushing her, taking into account factors such as the initial position, velocity, and acceleration.

## 2. How does this model work?

This model uses a mathematical equation, such as a sine or cosine function, to represent the displacement of the girl on the swing over time. The initial position, velocity, and acceleration are used as parameters to determine the shape and characteristics of the function.

## 3. What are the applications of this model?

This model can be used to study the motion of objects in a swinging motion, such as a pendulum or a child on a swing. It can also be used to analyze the effects of different forces on the motion of the girl on the swing, such as a change in the pushing force or the addition of friction.

## 4. How accurate is this model?

The accuracy of this model depends on the accuracy of the initial parameters and the simplicity of the function used. In real-life situations, there may be other factors that affect the motion of the girl on the swing, so this model should be used as a simplified representation.

## 5. How can this model be improved?

This model can be improved by incorporating more complex mathematical functions or by including other factors that may affect the motion, such as air resistance or the weight of the girl. It can also be improved by adjusting the initial parameters to better fit real-life scenarios.

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