A particle is projected at an angle of 30° with ground with speed 40 m/s. The speed of the particle after 2 s is (use g=10 ms-2)
A particle is projected at an angle of 30° with ground with speed 40 m/s. The speed of the particle after 2 s is (use g=10 ms-2)
Solution and Explanation
Initial velocity \((u) = 40 \;m/s\)
Angle of projection \((θ) = 30°\)
Acceleration due to gravity \((g) = 10 \;m/s²\)
Horizontal component \((u_x) = u\) \(\times\) \(cos(θ)\)
Vertical component \((u_y) = u \times sin(θ)\)
Horizontal component \((u_x) = 40 \times cos(30°) ≈ 34.64 \;m/s \)
Vertical component \((u_y) = 40 \times sin(30°) ≈ 20 \;m/s\)
velocity after \(2\) seconds: \(v_y = u_y + gt\)
Substituting the known values: \(v_y = 20 + (10 \times 2) = 20 + 20 = 40 \;m/s\)
Now, to find the resultant velocity after 2 seconds, we can use Pythagoras' theorem:
\(v = \sqrt{(v_x² + v_y²)}\)
\(v = \sqrt{((34.64)²} + (40)²) v ≈ \sqrt{(1199.13 + 1600) }≈ \sqrt{(2799.13)} ≈ 52.92 \;m/s\)
So, the speed of the particle after \(2\) seconds is approximately \(52.92\) \(m/s\).
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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 earth’s 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 – x = Vx0t
- Horizontal Velocity – Vx = Vx0
- Vertical Distance, y – Vy0t – ½ gt2
- Vertical Velocity, Vy – Vy0 – gt