Motion in plane practice · Medium

Option D (Zero) is correct because the vector projection onto the y-axis depends only on the y-component of the vector, which is zero in this case.

Option B (If both assertion and reason are true but reason is not the correct explanation of assertion) is correct because while both statements are true, the reason doesn't directly explain why we can treat the sum of two vectors as the difference of two vectors.

Option A (Statement-1 and Statement-2 are true, Statement-2 is the correct explanation for Statement-1) is correct because the constant acceleration due to gravity results in the same change in velocity over equal time intervals.

Option A (If both Assertion & Reason are True & the Reason is a correct explanation of the Assertion) is correct because the given position vector leads to a parabolic trajectory, and the reason correctly derives the parabolic equation.

Option C (${A}$ and ${B}$ acts in the opposite direction) is correct because the dot product is negative when the angle between the vectors is 180 degrees.

Option B (at each point of flight horizontal component of momentum remains constant) is correct because in projectile motion, the horizontal component of velocity remains constant, and therefore the horizontal component of momentum also remains constant.

Option D (Zero) is correct because at the highest point of a projectile launched at 90 degrees, the horizontal velocity is zero, resulting in zero kinetic energy.

Option D (If Assertion is false but Reason is true) is correct because the angle between $({A} \times {B})$ and $({B} \times {A})$ is $180^{\circ}$, making the cross product zero, and the reason correctly states that they are anti-parallel.

Option A ($2(\vec{b} \times \vec{a})$) is correct because expanding the cross product results in $2(\vec{b} \times \vec{a})$.

Option C (At the top most point) is correct because at the highest point, the vertical velocity is zero, and the acceleration due to gravity is vertically downwards, making them perpendicular.

Option D (Adding a scalar to a vector of the same dimensions) is correct because scalars and vectors are fundamentally different mathematical objects and cannot be directly added.

Option A is correct because the time it takes for an object to fall vertically depends only on the height and gravity, and both particles are dropped from the same height with the same gravitational acceleration.

Option C ($\sqrt{\frac{8 H}{g}}$) is correct because the time of flight is calculated using the maximum height and the acceleration due to gravity.

Option B ($\hat{i}-\hat{j}$) is correct because it represents the direction of the net acceleration (and therefore the force) at t=3s, considering both x and y components.

Option C is correct. By resolving the vectors and using trigonometric relationships, the magnitudes of the two vectors are found to be 6 units and 10 units.

Option D ($0^{\circ}$) is correct because the dot product magnitude equals the product of the magnitudes only when the angle between the vectors is 0 degrees.

Option C ($\frac{u^2}{2 g}$) is correct because the sum of maximum heights for complementary angles of projection equals $\frac{u^2}{2g}$.

Option B ($A$ is parallel to $B$) is correct because vector $B$ is a scalar multiple of vector $A$, indicating they are parallel.

The correct answer is A ($10 \hat{i}+10 \sqrt{2} \hat{j}+10 \hat{k}$) because it correctly represents the velocity vector in rectangular form using direction cosines.

The correct answer is B (${30^0}$) because the angle of projection is found by relating the range and maximum height using the given condition and trigonometric identities.

Option D ($\frac{\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}}{\sqrt{3}}$) is correct because it represents the unit vector along the diagonal of a cube with sides along the coordinate axes.

Option A (2) is correct because comparing the given equation with the standard trajectory equation allows us to determine the initial velocity.

Option B ($2 \sqrt{10} \mathrm{~m} / \mathrm{s}$) is correct because it is derived by comparing the given trajectory equation with the standard projectile motion equation and solving for the initial velocity.

Option A (perpendicular unit vectors) is correct because if the dot product of two vectors is zero, they are perpendicular, and if their cross product is 1, their magnitudes must be 1.

Option C (3) is correct because, for the vectors to be perpendicular, their dot product must be zero, which leads to a = 3.

Option B (7.5) is correct because the time taken in still water is calculated using the boat's speed and the river's speed, resulting in 7.5 hours.

Option C (2.5) is correct because it is the acceleration of the rocket calculated using the formula $a = \frac{v}{M} \frac{dM}{dt}$.

Option C ($90^\circ$) is correct because at the highest point, the velocity is horizontal and the acceleration due to gravity is vertical.

Option D (u/2) is correct because the horizontal component of velocity remains constant throughout projectile motion and is equal to $u \cos{\theta}$.

Option A ($2 \sqrt{3}$ m) is correct because the magnitude of $\vec{A}$ is calculated using the y-component and the given angle.