A toy rocket engine is securely fastened to a large puck that can glide with negligible friction over a horizontal surface, taken as the xy plane. The 5.60-kg puck has a velocity of 1.40î m/s at one instant. Eight seconds later, its velocity is (6.00î + 4.0ĵ) m/s.

(a) Assuming the rocket engine exerts a constant horizontal force, find the components of the force.
(___î + ___ĵ)N

(b) Find its magnitude.
___N

In the x direction

change in x speed = 6-1.4 = 4.6 m/s
Newton #2 is Force = rate of change of momentum
so Fx = 5.60 kg * 4.6 m/s / 8s = 3.22 Newtons
and
Fy = 5.60 * (4-0)/8 = 2.8 N
F = Fx i + Fy j

magnitude = sqrt (Fx^2+Fy^2)

thank you so much, that really helped!

wow thanks

(a) Well, let's do some calculations to find the components of the force.

Using the given information, we can calculate the change in velocity of the puck:

Δv = (6.00î + 4.0ĵ) m/s - 1.40î m/s
= 4.60î + 4.0ĵ m/s

Now, we can use the formula for force to find the components of the force:

Δp = FΔt

where Δp is the change in momentum, F is the force, and Δt is the time interval.

The mass of the puck is given as 5.60 kg, so we can write the formula as:

(5.60 kg) * (4.60î + 4.0ĵ m/s) = F * 8 s

Now let's solve for the components of the force:

Fî = (5.60 kg * 4.60 m/s) / 8 s
= 3.22 N

Fĵ = (5.60 kg * 4.0 m/s) / 8 s
= 2.80 N

Therefore, the components of the force are (3.22î + 2.80ĵ) N.

(b) To find the magnitude of the force, we can use the Pythagorean theorem:

|F| = √((Fî)^2 + (Fĵ)^2)

|F| = √((3.22^2) + (2.80^2))
= √(10.41 + 7.84)
= √18.25
≈ 4.27 N

So, the magnitude of the force is approximately 4.27 N.

To solve this problem, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. In this case, the object we are considering is the puck, and we need to find the force exerted by the rocket engine.

Let's start with part (a) and find the components of the force:

We know the mass of the puck, which is 5.60 kg. Let's call the force exerted by the rocket engine F, and the acceleration of the puck a.

Using Newton's second law, we can write the equation as:
F = m * a

We can also use the relationship between force, mass, and acceleration to rewrite the equation as:
F = m * Δv / Δt

Where Δv is the change in velocity and Δt is the change in time.

In this case, the change in velocity is given by:
Δv = (6î + 4ĵ) m/s - 1.40î m/s

Substituting the values, we have:
Δv = (6î + 4ĵ) m/s - 1.40î m/s
Δv = 4.60î + 4ĵ m/s

The change in time is given as 8 seconds.

Now, let's find the acceleration:
a = Δv / Δt
a = (4.60î + 4ĵ m/s) / 8 s

To calculate the acceleration:
a = (4.60î + 4ĵ m/s) / 8 s
a = (0.575î + 0.5ĵ) m/s^2

Now we have the acceleration of the puck. To find the force exerted by the rocket engine F, we can substitute this acceleration value into the equation:
F = m * a
F = 5.60 kg * (0.575î + 0.5ĵ) m/s^2

Simplifying this equation, we get:
F = (3.22î + 2.8ĵ) N

Therefore, the components of the force exerted by the rocket engine are (3.22î + 2.8ĵ) N.

Moving on to part (b), we need to find the magnitude of the force. The magnitude of a vector is given by the square root of the sum of the squares of its components.

Let's calculate the magnitude of the force F:
|F| = sqrt(3.22^2 + 2.8^2) N

Using a calculator:
|F| = sqrt(10.4084 + 7.84) N
|F| = sqrt(18.2484) N
|F| = 4.27 N (approximately)

Therefore, the magnitude of the force exerted by the rocket engine is 4.27 N.