Mechanics 1 - general motion Qs

hiya

Could someone help on these >< :frown:

~~

1. A particle moves on the x-axis. It's displacement, x m, from the origin O is given by: x = 3t² - 3t + 2, where t is the time in seconds.

How far is the particle from O when it is instantaneously at rest?

~~

2. In this question, i and j are the standard unit vectors in the Ox and Oy directions.

An object has initial position (2i - j)m and velocity (-i + 4j)ms^-1.
It has a costant acceleration of (2i + 5j)ms^-2.

Calculate the object's velocity and position after four seconds.

~~

3. A racing car starts off down a straight section of the track towards the first corner. Its speed, vms^-1, is modelled for the first four seconds of its motion by: v = t³ - 9t² + 24t, 0 < t < 4.

Find an expression for the distance travelled by the car in the first t seconds.

~~

4. The position vector, r, of a particle at time t is given by: r = t² + (5t - 2t²)j,
where i and j are the standard unit vectors, lengths are in metres and time is in seconds.

Find an expression for the acceleration of the particle.

~~

im really hopeless at these :frown:

thx

Reply 1

ImperceptibleNinja

"Q1. A particle moves on the x-axis. It's displacement, x m, from the origin O is given by: x = 3t^2 - 3t + 2, where t is the time in seconds.

How far is the particle from O when it is instantaneously at rest?"

dx/dt = 6t - 3 = 0 --> t = 1/2
x = 3(1/2)^2 - 3(1/2) + 2 = 5/4m

"3. A racing car starts off down a straight section of the track towards the first corner. Its speed, vms-1, is modelled for the first four seconds of its motion by: v = t^3 - 9t^2 + 24t, 0 < t < 4.

Find an expression for the distance travelled by the car in the first t seconds."

s = integral of v
s = (t^4)/4 - 3t^3 + 12t^2 + c
at s=0, t=0
0 = c
s = (t^4)/4 - 3t^3 + 12t^2

I'll see if i can come up with anything for the vectors, but don't get too hopeful on that :smile:

Reply 2

*bobo*

2) a=(v-u)/t
(2i+5j)=(v-(-1+4j))/4
v=(9i+16j)m/s

Reply 3

WigglesXW

hiya

thx for the help!!

Could someone have a go at 4 for me? i really am crap at these :frown:

thx

Reply 4

Jaswarbrick

4. The position vector, r, of a particle at time t is given by: r = t^2 i + (5t - 2t^2)j,
where i and j are the standard unit vectors, lengths are in metres and time is in seconds.

Find an expression for the acceleration of the particle.

I think you just have to differentiate twice to get acceleration.

r = t^2 i + (5t - 2t^2) j
v = 2t i + (5 - 4t) j
a = 2 i - 4 j

Hope this helps.

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