A missile is fired from the ground level rises x meters vertically upwards in t sec, where \(x=100t-\frac{25}{2}t^2\). the maximum height reached is
- 100m
- 300m
- 200m
- 125m
The Correct Option is D
Solution and Explanation
The correct answer is option (D): 125m
Given upward displacement in time t second
\(x=100t\frac{25}{2}t^2\)
Initial velocity = \(\frac{dx}{dt}=100-25t=100-25\times0=100\,m/s\)
At the maximum height velocity \(\frac{dx}{dt}=0\)
\(100-25t=0\Rightarrow t=4\)
Maximum height reached \(x=100\times4-\frac{25}{2}\times16=400-200=200m\)
Hence the maximum height reached is 200m.
The velocity of the missile, when it reaches the ground, is \((\frac{dx}{dt})_{t=8}=100-25\times 8=-100m/s.\)
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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