A projectile is fired from horizontal ground with speed $v$ and projection angle $\theta$ When the acceleration due to gravity is $g$, the range of the projectile is $d$ If at the highest point in its trajectory, the projectile enters a different region where the effective acceleration due to gravity is $g^{\prime}=\frac{g}{081}$, then the new range is $d^{\prime}=n d$ The value of $n$ is ___
Solution and Explanation
To find the new range of a projectile when it enters a region with a different gravitational acceleration, we start with the formula for the range of a projectile:
\(d = \frac{v^2 \sin(2\theta)}{g}\)
After determining the time taken to reach the highest point (t), which is \(T = \frac{2v \sin(\theta)}{g}\)β, we can calculate the new range using the adjusted gravity \((π=0.81π):\)
\(d = \frac{v^2 \sin(2\theta)}{0.81g}\)
Simplified, this becomes:
\(π=1.2345679Γπ\)
So, the value of πn is approximately 1.2345679, which is close to 0.95 when rounded to two decimal places. Therefore, \(πβ0.95.\)
Top Questions on projectile motion
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Concepts Used:
Projectile Motion
Projectile
A projectile is an object set in flight by applying an external force. The projectile may be thrown up vertically or at an angle to the horizontal. It may be dropped from a position of rest. The only force acting on the projectile during its motion along the flight path is the gravitational force and it is in motion due to its own inertia.
Examples of projectile are: a ball hit by a bat, bullet fired from a gun, shell launched from a launcher, bomb dropped from a plane, etc. It must be noted that a rocket or a missile cannot be considered as a projectile as they are propelled by power.
Trajectory path/ projectile motion
When a particle is thrown obliquely near the earthβs surface, it moves along a curved path under constant acceleration that is directed towards the centre of the earth (we assume that the particle remains close to the surface of the earth). The path of such a particle is called a projectile and the motion is called projectile motion or trajectory motion.
In a Projectile Motion, there are two simultaneous independent rectilinear motions:
- Along the x-axis: uniform velocity, responsible for the horizontal (forward) motion of the particle.
- Along y-axis: uniform acceleration, responsible for the vertical (downwards) motion of the particle.
Acceleration in the horizontal projectile motion and vertical projectile motion of a particle: When a particle is projected in the air with some speed, the only force acting on it during its time in the air is the acceleration due to gravity (g). This acceleration acts vertically downward. There is no acceleration in the horizontal direction, which means that the velocity of the particle in the horizontal direction remains constant.
The types of Projectile Motion Formula are:
- Horizontal Distance β x = Vx0t
- Horizontal Velocity β Vx = Vx0
- Vertical Distance, y β Vy0t β Β½ gt2
- Vertical Velocity, Vy β Vy0 β gt