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MCQEasyJEE 2023Projectile Motion

JEE Physics 2023 Question with Solution

Two projectiles A and B are thrown with initial velocities of 40 m/s40 \, \text{m/s} and 60 m/s60 \, \text{m/s} at angles 30∘30^\circ and 60∘60^\circ with the horizontal respectively. The ratio of their ranges is (g=10 m/s2g = 10 \, \text{m/s}^2):

  • A

    2:32 : \sqrt{3}

  • B

    3:2\sqrt{3} : 2

  • C

    4:94 : 9

  • D

    1:11 : 1

Answer

Correct answer:C

Step-by-step solution

Standard Method

Given: Projectile A has initial speed u1=40 m/su_1 = 40 \, \text{m/s} and angle θ1=30∘\theta_1 = 30^\circ. Projectile B has initial speed u2=60 m/su_2 = 60 \, \text{m/s} and angle θ2=60∘\theta_2 = 60^\circ. Also, g=10 m/s2g = 10 \, \text{m/s}^2.

Find: The ratio of their horizontal ranges.

For projectile motion, the range is

R=u2sin⁡2θgR = \frac{u^2 \sin 2\theta}{g}

So, for two projectiles,

R1R2=u12u22⋅sin⁡2θ1sin⁡2θ2\frac{R_1}{R_2} = \frac{u_1^2}{u_2^2} \cdot \frac{\sin 2\theta_1}{\sin 2\theta_2}

Substitute the given values:

R1R2=402602⋅sin⁡60∘sin⁡120∘\frac{R_1}{R_2} = \frac{40^2}{60^2} \cdot \frac{\sin 60^\circ}{\sin 120^\circ}

Since sin⁡60∘=sin⁡120∘\sin 60^\circ = \sin 120^\circ, this becomes

R1R2=402602=16003600=49\frac{R_1}{R_2} = \frac{40^2}{60^2} = \frac{1600}{3600} = \frac{4}{9}

Therefore, the ratio of the ranges is 4:94 : 9. The correct option is C.

Direct Evaluation of Both Ranges

Given: u1=40 m/su_1 = 40 \, \text{m/s}, θ1=30∘\theta_1 = 30^\circ, u2=60 m/su_2 = 60 \, \text{m/s}, θ2=60∘\theta_2 = 60^\circ, and g=10 m/s2g = 10 \, \text{m/s}^2.

Find: The ratio R1:R2R_1 : R_2.

For the first projectile,

R1=402sin⁡60∘10R_1 = \frac{40^2 \sin 60^\circ}{10}

For the second projectile,

R2=602sin⁡120∘10R_2 = \frac{60^2 \sin 120^\circ}{10}

Now compare them:

R1:R2=402sin⁡60∘:602sin⁡120∘R_1 : R_2 = 40^2 \sin 60^\circ : 60^2 \sin 120^\circ

Using sin⁡120∘=sin⁡60∘\sin 120^\circ = \sin 60^\circ, we get

R1:R2=402:602=1600:3600=4:9R_1 : R_2 = 40^2 : 60^2 = 1600 : 3600 = 4 : 9

Hence, the required ratio is 4:94 : 9.

Common mistakes

  • Using sin⁡θ\sin \theta instead of sin⁡2θ\sin 2\theta in the range formula is incorrect because the horizontal range for projectile motion is R=u2sin⁡2θgR = \frac{u^2 \sin 2\theta}{g}. Always use the double-angle term.

  • Taking sin⁡120∘\sin 120^\circ as different from sin⁡60∘\sin 60^\circ in value leads to a wrong ratio. Since sin⁡120∘=sin⁡60∘\sin 120^\circ = \sin 60^\circ, these factors cancel.

  • Comparing the speeds directly as 40:6040:60 instead of squaring them is wrong because the range depends on u2u^2, not on uu. First square the speeds, then form the ratio.

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