Question:
A ball is dropped from a platform $19.6\,m$ high. Its position function is -
A ball is dropped from a platform $19.6\,m$ high. Its position function is -
Updated On: Sep 30, 2024
- $x = - 4.9t^2 + 19.6 (0 \le t \le 1)$
- $x = - 4.9t^2 + 19.6 (0 \le t \le 2)$
- $x = - 9.8t^2 + 19.6 (0 \le t \le 2)$
- $x = - 4.9t^2 - 19.6 (0 \le t \le 2)$
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The Correct Option is B
Solution and Explanation
We have, $a=\frac{d^{2} x}{d t^{2}}=-9.8$
The initial conditions are $x(0)=19.6$ and $v(0)=0$
So, $v=\frac{d x}{d t}=-9.8 t+v(0)=-9.8 t$
$\therefore x=-4.9 t^{2}+x(0)=-4.9 t^{2}+19.6$
Now, the domain of the function is restricted since the ball hits the ground after a certain time. To find this time we set $x=0$ and solve for $t$.
$0=4.9 t^{2}+19.6 \Rightarrow t=2$
The initial conditions are $x(0)=19.6$ and $v(0)=0$
So, $v=\frac{d x}{d t}=-9.8 t+v(0)=-9.8 t$
$\therefore x=-4.9 t^{2}+x(0)=-4.9 t^{2}+19.6$
Now, the domain of the function is restricted since the ball hits the ground after a certain time. To find this time we set $x=0$ and solve for $t$.
$0=4.9 t^{2}+19.6 \Rightarrow t=2$
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Concepts Used:
Differential Equations
A differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on.
Orders of a Differential Equation
First Order Differential Equation
The first-order differential equation has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’
Second-Order Differential Equation
The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.
Types of Differential Equations
Differential equations can be divided into several types namely
- Ordinary Differential Equations
- Partial Differential Equations
- Linear Differential Equations
- Nonlinear differential equations
- Homogeneous Differential Equations
- Nonhomogeneous Differential Equations