Projectile Motion and Range

Projectile motion refers to the motion of an object that is launched into the air and moves along a curved path under the influence of gravity. The range of a projectile is the horizontal distance covered by the object before it hits the ground. In this case, we are given that the projectile is thrown at an angle of 30 degrees to the horizontal. We need to determine the range of initial velocity (u) for which the range falls between 40m and 50m.

Calculating Range

The range of a projectile can be calculated using the formula:

Range = (u^2 * sin(2θ)) / g

where u is the initial velocity, θ is the angle of projection, and g is the acceleration due to gravity.

Substituting the given values, we have:

Range = (u^2 * sin(60))/10

Simplifying further, we have:

Range = (u^2 * √3/2)/10

Range = (u^2 * √3)/20

Range Between 40m and 50m

To find the range of initial velocity (u) that falls between 40m and 50m, we can set up an inequality:

40 < (u^2="" *="" √3)/20="" />< />

Multiplying both sides of the inequality by 20, we get:

800 < u^2="" *="" √3="" />< />

Dividing both sides of the inequality by √3, we have:

800/√3 < u^2="" />< />

Taking the square root of both sides, we get:

√(800/√3) < u="" />< />

Simplifying further, we have:

√(800√3)/3 < u="" />< />

Approximating the values, we find:

17.38 < u="" />< />

Therefore, the range of initial velocity (u) should be between approximately 17.38 m/s and 20.41 m/s in order for the range to fall between 40m and 50m.

Conclusion

In projectile motion, the range of a projectile can be calculated using the formula Range = (u^2 * sin(2θ)) / g. By substituting the given angle of projection and acceleration due to gravity, we can determine the range in terms of the initial velocity (u). In this case, the range is between 40m and 50m. By setting up an inequality and solving for u, we find that the initial velocity should be between approximately 17.38 m/s and 20.41 m/s.