Question

An object of mass \(3 \mathrm{~kg}\) is moving with a velocity of \(5 \mathrm{~m} / \mathrm{s}\) along a straight path. If a force of \(12 \mathrm{~N}\) is applied for \(3 \mathrm{sec}\) on the object in a perpendicular to its direction of motion. The magnitude of velocity of the particle at the end of \(3 \mathrm{sec}\) is \(\mathrm{m} / \mathrm{s}\). (A) (B) 12 (C) 13 (D) 4

Short answer

The magnitude of the final velocity of the object after the force is applied for 3 seconds is \(13 \: \mathrm{m/s}\), which corresponds to option (C).

Step-by-step solution

Identify given values

Here's a summary of the given information: - Mass of object (m): 3 kg - Initial velocity (v1): 5 m/s (along a straight path) - Force applied (F): 12 N (perpendicular to the direction of motion) - Time force is applied (t): 3 s

Determine acceleration due to perpendicular force

Using Newton's second law, we can express the acceleration due to the force (a2) as follows: \[a_2 = \frac{F}{m}\] where F = 12 N (force applied) m = 3 kg (mass of the object) Now we can find the acceleration: \[a_2 = \frac{12}{3} = 4 \: \mathrm{m/s^2}\]

Calculate the velocity resulting from the perpendicular force

We can use the equation for calculating final velocity (v2) after a given time (t) and acceleration (a2): \[v_2 = a_2 \times t \] Substituting the given values: \[v_2 = 4 \: \mathrm{m/s^2} \times 3 \: \mathrm{s} = 12 \: \mathrm{m/s}\]

Combine the initial velocity and the velocity due to the force

Since the initial velocity (v1) and the velocity due to the force (v2) are perpendicular to each other, we can use the Pythagorean theorem to find the magnitude of the final velocity (v): \[v = \sqrt{v_1^2 + v_2^2}\] Substituting the given values: \[v = \sqrt{(5 \: \mathrm{m/s})^2 + (12 \: \mathrm{m/s})^2} = \sqrt{169} = 13 \: \mathrm{m/s}\] The magnitude of the final velocity of the object after the force is applied for 3 seconds is 13 m/s, which corresponds to option (C).

Key concepts

Kinematics

Kinematics is a branch of physics that deals with the motion of objects without considering the forces that cause the motion. It's all about description, focusing on parameters like displacement, velocity, and acceleration.

- **Displacement**: How far an object has moved from its initial position.- **Velocity**: The rate of change of displacement, indicating direction. It's different from speed, which is just a scalar quantity.- **Acceleration**: The rate of change of velocity over time.

In the context of our exercise, the object starts with an initial velocity (\(v_1 = 5 \, \mathrm{m/s}\)) along a straight line. This involves kinematic equations, especially when force is applied as Newton's second law comes into play. Understanding how individual velocities can result in different paths when forces are applied is key in kinematics.

Vector Addition

Vectors are quantities that have both magnitude and direction. When dealing with forces and velocities, vector addition becomes crucial, especially when they are not aligned.

In the problem, the force is applied perpendicular to the initial velocity. This means the velocity change due to this force is also perpendicular. To find the resultant velocity after the force is applied:

• Calculate the change in velocity due to the force using \(v_2 = a_2 \times t\), giving \(v_2 = 12 \, \mathrm{m/s}\).
• The initial velocity vector is \(5 \, \mathrm{m/s}\), and the perpendicular velocity change is \(12 \, \mathrm{m/s}\).
• Apply the Pythagorean theorem: \(v = \sqrt{v_1^2 + v_2^2}\), resulting in \(13 \, \mathrm{m/s}\).

Breaking down vectors into components can help visualize their effect, especially when dealing with motion in two dimensions.

Perpendicular Force Effect

When a force is applied perpendicular to the direction of motion, it doesn't change the speed of the object along the original direction, but it changes the overall path.

Here's what happens:
- The object continues its motion with the initial velocity in its original direction. - The perpendicular force creates an additional velocity in a direction at right angles to the initial velocity.

As a result, the object's path becomes a diagonal in the resulting vector decomposition. This is because the perpendicular force causes a change in direction but not in the parallel component of movement.

• Think about how a frisbee moves when you flick your wrist – it goes sideways in addition to its forward path. That's a similar effect!